1. Introduction

Quantum theory departs sharply from classical intuitions about physical reality, most notably in its treatment of correlations. At the heart of this departure lies a structural question: is it possible to assign a globally consistent, context-independent probability or value assignment to all observables of a physical system? In classical theories, such global assignments exist and underpin realistic descriptions based on predetermined properties. In quantum theory, by contrast, this possibility is fundamentally obstructed. This obstruction is most clearly articulated in foundational no-go theorems. Gleason’s theorem^[1] shows that for Hilbert spaces of dimension d ≥ 3, any probability measure over projectors satisfying natural consistency conditions must coincide with the Born rule^[2], thereby ruling out noncontextual global probability assignments. The Kochen-Specker (KS) theorem^[3] provides a finite and combinatorial manifestation of the same obstruction, demonstrating that even for finite sets of observables it is impossible to assign predetermined, context-independent values consistent with quantum predictions. Motivated by the idealized assumptions underlying the traditional KS framework, Spekkens later introduced a generalized operational notion of contextuality applicable to arbitrary operational theories^[4], identifying contextuality with the impossibility of a context-independent ontological model for preparations, transformations, and measurements. Together, these results establish contextuality as an intrinsic nonclassical feature of quantum theory.

The same structural obstruction manifests operationally in different physical scenarios. In spatially separated multipartite systems it gives rise to Bell nonlocality^[5]. Bell’s theorem^[6] proves that no theory satisfying local causality can reproduce all quantum correlations while respecting the no-signaling constraints imposed by relativistic causality. In a standard Bell scenario, independent measurement choices performed at distant locations generate joint statistics that cannot be decomposed into local hidden variables^[5]. Since its experimental confirmation^[7-12], Bell nonlocality has evolved from a foundational paradox into a central operational concept in quantum information science^[13-15]. In contrast, contextuality, which has been identified as a key resource for speeding up quantum computation^[16,17], can manifest within a single quantum system through the compatibility structure of measurements^[18]. Operationally, contextual scenarios are governed by the no-disturbance condition, requiring that the marginal statistics of a measurement remain invariant under the choice of jointly performed compatible observables.

Surprisingly, despite nonlocality and contextuality differing in their physical realization, both phenomena can be unified as manifestations of structural obstructions to noncontextual explanations in the sheaf-theoretic framework^[19]. This perspective gives rise to a three-level hierarchy of nonclassical correlations, which was given a unified and precise formulation in the sheaf-theoretic framework developed by Abramsky and Brandenburger^[19]. At the weakest level lies probabilistic nonlocality, exhibited by violations of Bell-type inequalities derived under local hidden-variable (LHV) assumptions^[5]. Historically, this level originates from Bell’s theorem and its subsequent experimental tests^[6-9], which established nonlocality as a statistical incompatibility between quantum predictions and classical realism under relativistic causality, and initiated the modern study^[10-12]. The intermediate level corresponds to possibilistic (or logical) nonlocality, exemplified by Hardy-type paradoxes^[20-22]. Here the contradiction is not merely statistical but logical: certain combinations of events are impossible in classical models yet occur with nonzero probability in quantum theory. At the highest level is strong contextuality, which exhibits deterministic, combinatorial contradictions. These structures appeared in many models, first in the Greenberger-Horne-Zeilinger (GHZ) construction^[23], and were subsequently reformulated by Mermin into compact algebraic proofs^[24-26], providing both multipartite Bell contradictions and state-independent KS demonstrations without inequalities. Within the sheaf theory^[19], KS contextuality^[3] is naturally positioned at the end of the hierarchy, reflecting a maximal obstruction to global consistent value assignments.

Complementing the sheaf-theoretic perspective, graph-theoretic methods^[27] provide an operationally meaningful characterization of contextuality and nonlocality by encoding exclusivity relations between measurement events into combinatorial structures. Graph invariants such as the independence number, Lovász number, and fractional packing number delimit classical, quantum, and general probabilistic correlations, respectively. These tools yield experimentally testable inequalities and have proven particularly powerful for connecting foundational structures with concrete implementations. Indeed, early investigations into the logical foundations of quantum mechanics, including KS and locality theorems^[28-31] and related no-go results^[32,33], had already anticipated this deep connection between strong contextuality and nonlocal correlations, which later found related ideas in GHZ-like constructions for two-qubit systems^[34], the Peres-Mermin square^[26,35,36], all-versus-nothing proofs^[37], and formulations in terms of nonlocal games^[38] and quantum pseudotelepathy^[39,40]. Building on these foundational insights, recent developments have further clarified the theoretical and operational connections between contextuality and nonlocality^[41-43]. Collectively, these results indicate that contextuality and nonlocality are not isolated phenomena but manifestations of structural principles governing quantum correlations. Figure 1 illustrates this correspondence by comparing the Bell Clauser-Horne-Shimony-Holt (CHSH) scenario^[7], the simplest Bell scenario capable of revealing quantum nonlocality, with the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario^[44], the simplest contextuality scenario capable of revealing state-dependent contextuality (SD-C) in a single qutrit. Despite their different physical realizations, both scenarios exhibit the same fundamental contextual structure: observables are shared among multiple measurement contexts, and any noncontextual hidden-variable model must assign outcomes consistently across these contexts.

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Figure 1. Comparison between nonlocality and contextuality scenarios from the perspective of measurement contexts. (a) Bell-CHSH scenario, the simplest Bell scenario used to demonstrate nonlocal correlations between two spatially separated qubits. Adapted with permission from reference^[7]. Copyright © 1969 American Physical Society. For clarity, only two representative contexts are explicitly shown, the same local observable A₁ is performed in two different contexts, {A₁, B₁} and {A₁, B₂}. The remaining CHSH contexts, {A₂, B₁} and {A₂, B₂}, are omitted for simplicity. Here Alice’s and Bob’s measurements are performed in spacelike separated regions, measurements follow from the locality assumption; (b) KCBS contextuality scenario, the simplest contextuality scenario involving a single qutrit and capable of revealing state-dependent contextuality. Adapted with permission from reference^[44]. Copyright © 2008 American Physical Society. For clarity, only the representative contexts {C₁, C₂} and {C₂, C₃} and {C₅, C₁} are explicitly shown. The remaining KCBS contexts, {C₃, C₄} and {C₄, C₅} are omitted by the ellipsis. In the KCBS cycle, each observable participates in two different compatible contexts. Unlike the Bell scenario, consistency across contexts is established through compatibility and the nondisturbance condition rather than spacelike separation. In contextuality experiments, additional measurements are often required to assess compatibility and nondisturbance. Adapted with permission from reference^[45]. Copyright © 2013 American Physical Society; reference^[46]. Copyright © 2015 Springer Nature. The measurement of C_i is therefore repeated either alone or together with C_{i+1(mod 5)} to quantify possible measurement disturbance and experimental imperfections. Adapted with permission from reference^[47]. Copyright © 2009 Springer Nature; reference^[48]. Copyright © 2010 American Physical Society. Consistency between the initial and repeated outcomes, represented schematically by the yellow light on the device, provides an operational test of measurement compatibility. CHSH: Clauser-Horne-Shimony-Holt; KCBS: Klyachko-Can-Binicioğlu-Shumovsky.

From an experimental standpoint, realizing and testing these abstract structures poses significant challenges. Concepts such as compatibility, exclusivity, and global consistency must be translated into experimentally accessible protocols, while accounting for imperfections including finite detection efficiency, noise, and limited measurement sharpness. Among available platforms, photonic systems—particularly structured-light and orbital-angular-momentum (OAM) platforms—offer exceptional controllability, access to high-dimensional Hilbert spaces, and flexible measurement design, making them especially well suited for testing contextuality and nonlocality within a common experimental architecture. Photonic implementations have therefore played a central role in transforming foundational no-go theorems into quantitatively testable and technologically relevant phenomena.

In this Review, we provide a comprehensive overview of the conceptual and experimental connections between quantum contextuality and nonlocality. Section 2 introduces the sheaf-theoretic framework and its logical structure. Section 3 presents the graph-theoretic approach and its operational implications. Section 4 surveys photonic experimental realizations that probe these connections. We conclude by outlining open problems and future directions, emphasizing how a deeper structural understanding of contextuality and nonlocality may inform both fundamental research and the systematic development of quantum technologies.

2. Sheaf-Theoretic Approach

Understanding quantum correlations in a unified and conceptually transparent way requires a framework that treats contextuality and nonlocality on equal footing. These phenomena arise in distinct operational settings: nonlocality involves spatially separated measurements, whereas contextuality involves compatible measurements on a single system. Nevertheless, both originate from the same structural limitation of classical models, which cannot assign globally consistent hidden variables that reproduce the observed correlations.

A powerful unifying framework developed by Abramsky and Brandenburger^[19] employs sheaf theory to precisely describe empirical models, in which contextuality and nonlocality are naturally unified and characterized as obstructions to constructing globally consistent descriptions of measurement outcomes. Built on presheaves defined over measurement contexts, this approach reveals that both phenomena arise from a single structural limitation: the impossibility of assigning outcome values that reproduce all locally observed statistics. From a physical perspective, cohomological methods can be viewed as tools for determining whether locally consistent outcome assignments can be extended to a globally consistent description. A nontrivial cohomological obstruction signifies that such an extension is impossible, thereby witnessing contextuality. The significance of this approach lies in its ability to detect contextuality through the global structure of a measurement scenario rather than through the violation of a particular inequality. Within this framework, the central distinction lies between local consistency, which requires agreement among overlapping contexts, and global consistency, which demands the existence of a joint assignment for all measurements. The failure to obtain such a global assignment reflects the limitation of classical hidden-variable models, arising from spatial separation in the case of nonlocality and from incompatible measurements in the case of contextuality.

This section presents a survey of the sheaf-theoretic framework for contextuality and nonlocality. We begin by presenting its basic formal structure and then explain how both phenomena arise as instances of a general obstruction to constructing globally consistent descriptions of measurement statistics. Next, we apply the powerful tools of presheaf cohomology to witness and characterise non-locality and contextuality. We conclude by discussing implications for quantum foundations and connections to broader mathematical structures, such as cohomology and logic. An advantage of the sheaf-theoretic framework is that it comes equipped with a particular representation that provides a powerful means of reasoning about empirical models.

2.1 The Sheaf-theoretic framework

To study contextuality and nonlocality in a unified way, we need a formalism that captures two essential aspects of quantum experiments: which measurements can be performed together, and how the outcomes are distributed. Operationally, a single run of such an experiment consists of selecting a set of compatible measurements, performing them on a physical system, and observing the resulting outcomes. Each run yields an event, defined by the chosen measurements and their outcomes. Repeated trials yield relative frequencies, which for each jointly performable set of measurements define a probability distribution over events. The collection of such distributions across all measurement contexts constitutes an empirical model. Formally, one begins by specifying a measurement scenario (X, O, M), where X is a finite set of measurements, O is a finite set of outcomes, and M is a family of measurement contexts C ⊆ X, representing sets of measurements that can be performed jointly. For each C ∈ M, one specifies a distribution on the events O^C. To organize a structural description of empirical models, we introduce the mathematical language of presheaf and sheaf. Let P(X) denote the poset of subsets of X, ordered by inclusion and regarded as a category in the usual way.

Definition 1. A presheaf on P(X) is a functor.

(1)$E:P(X{)}^{\mathrm{op}}\to Set$

For each subset C ⊆ X, the presheaf Ɛ assigns a set Ɛ(C), and for each inclusion C′ ⊆ C, a restriction map Ɛ(C)→Ɛ(C′). Functoriality ensures that restrictions compose consistently and that the identity inclusion induces the identity map. Elements of Ɛ(C) are called sections over C, which represent assignments of outcomes to the measurements in C. If X itself is included as an object, a section over X is referred to as a global section. In the present context, the most important example is the event presheaf, defined by Ɛ(C) := O^C, the set of all outcome assignments to the measurements in C. Restriction maps are given by marginalization, corresponding to forgetting outcomes of measurements not contained in the smaller context. A global section of this presheaf therefore corresponds to a deterministic assignment of outcomes to all measurements in X.

The concept of a sheaf refines that of a presheaf by imposing a consistency requirement that local data determine global data uniquely whenever they are mutually compatible.

Definition 2. A presheaf Ɛ on P(X) is a sheaf if for any family of subsets {C_i} with C = ∪_iC_i, and any family of sections s_i ∈ Ɛ(C_i) satisfying the compatibility condition.

(2)${s_i|}_{C_i\cap C_j}={s_j|}_{C_i\cap C_j}\forall i,j$

There exists a unique section s ∈ Ɛ(C) such that s|C_i = s_i for all i.

A useful intuition is that a presheaf assigns information to a poset such that the assignment on any element can be consistently restricted to lower elements. A sheaf further requires that assignments which exist and are locally compatible on all lower elements can be glued, or lifted, to an assignment on the element itself. Physically, this corresponds to the classical intuition that deterministic outcome assignments for overlapping sets of compatible measurements can always be extended consistently. Figure 2 illustrates the central idea underlying the sheaf-theoretic approach. Local sections represent outcome assignments within individual measurement contexts, while a global section corresponds to an assignment defined consistently over all measurements. When local sections agree on the measurements shared by overlapping contexts, the sheaf condition guarantees that they can be glued into a unique global section. Contextuality and nonlocality arise precisely when locally consistent empirical data fail to admit such a globally consistent assignment. The figure therefore highlights the fundamental distinction between local consistency and global consistency that underlies the sheaf-theoretic analysis of empirical models.

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Figure 2. Illustration of the sheaf-theoretic characterization of contextuality and nonlocality. Measurements are organized into overlapping contexts, here represented by C_A = A₁, A₂, A₃ and C_B = B₁, B₂, A₃, where the observable A₃ belongs to both contexts. For each context C, a local section s_C ∈ Ɛ(C) assigns outcomes to all measurements within that context. The central question is whether these local sections agree on the overlap C_A ∩ C_B and can therefore be glued into a global section s ∈ Ɛ(X). If the local sections are compatible on all overlaps, the sheaf condition guarantees the existence of a unique global assignment, corresponding to a noncontextual (or local) hidden-variable description. If no such global section exists, the empirical model is contextual; in Bell scenarios, this obstruction manifests as nonlocality. The figure illustrates the fundamental distinction between local consistency and global consistency that lies at the heart of the sheaf-theoretic approach.

The relevance of this structure to quantum contextuality is immediate. In KS scenarios^[3], it is possible to assign outcomes consistently within each measurement context, yet impossible to construct a single global assignment compatible with all contexts simultaneously. An intuitive geometric analogy is provided by the Penrose triangle in Figure 3, which admits locally consistent constructions but cannot be realized as a single global object. Such situations correspond precisely to families of local sections of the event presheaf that do not arise as restrictions of any global section. This perspective was first articulated in the language of presheaves by Isham and Butterfield^[50], laying the groundwork for the topos approach to quantum theory. While the topos framework incorporates substantial mathematical structure from operator algebras and quantum logic, the sheaf-theoretic approach adopted here is deliberately more operational and minimalist. Its aim is not to reformulate quantum theory itself, but rather to characterize and compare the constraints imposed on empirical data by different physical or hypothetical theories, using only experimentally accessible correlations.

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Figure 3. Interpreting “>” as “appears visually closer”, the Penrose triangle gives rise to locally consistent orderings that cannot be obtained as the restrictions of any global section. Adapted from reference^[49]. CC BY 4.0. A global section would correspond to a strict total order on {A, B, C}, which is incompatible with these visual assignments.

Example 1. For a triangle, as shown in Figure 3, label the edges by {A, B, C}, take as poset subsets of the edges labelled by inclusion, and define a presheaf Ɛ that assigns to each subset the set of all strict total orderings of its elements.

(3)$E\left(\right\{A,B\left\}\right)=\{A>B,B>A\}$

Restrictions arise in the obvious way. The triangle represents a family of sections.

(4)$\{s_{\{A,B\}}=B>A,s_{\{B,C\}}=C>B,s_{\{C,A\}}=A>C\}$

This cannot arise from restrictions of any global section s_{A,B}, which in this case would be a strict total order on {A, B, C}.

To describe empirical models quantitatively, we specify a probability distribution over the assignments Ɛ(C) for each maximal context C ∈ M. This can be achieved by composing Ɛ with the distribution functor D_R: Set→Set, which maps a set to the set of R-distributions over it, where R is a semiring. Standard probability distributions correspond to $R=R_{\ge 0}$, the non-negative reals, while other generalizations, such as ‘negative probabilities $(R=R)$^[51-54] or ‘possibilistic’ distributions ($R=B$, the Boolean semiring)^[55,56], can also be considered. The composition D_RƐ is a presheaf, with restrictions given by marginalization of distributions.

Definition 3. An empirical model e over a measurement scenario (X, O, M) is a family of distributions.

(5)${\{e_C\in D_{R}E(C\left)\right\}}_{C\in M}$

It assigns a distribution to each measurement context.

Not every such family represents a physically admissible model. Operational consistency requires that distributions associated with different contexts agree on their overlaps.

(6)${e_C|}_{C\cap C^{\prime }}={e_{C^{\prime }}|}_{C\cap C^{\prime }}\forall C,C^{\prime }\in M$

This compatibility condition is precisely the sheaf condition restricted to the measurement cover M. In multipartite Bell scenarios it coincides with the no-signaling constraint^[57], while in single-system contextuality scenarios it corresponds to the no-disturbance condition^[58]. Empirical models satisfying this condition are therefore referred to as compatible. The following example illustrates how an empirical model arising from quantum mechanics is represented and analyzed within the sheaf-theoretic framework.

Example 2. Quantum mechanics provides a central class of empirical models, constructed by specifying a quantum state together with a set of observables and evaluating the joint probability distributions of outcomes for each compatible measurement context. As a concrete illustration of this construction, we consider the standard CHSH scenario^[7]. Suppose that two parties, Alice and Bob, share the two-qubit Bell state.

(7)$|{\varphi }^{+}⟩=\frac{1}{\sqrt{2}}(|0{⟩}_A\otimes |0{⟩}_B+|1{⟩}_A\otimes |1{⟩}_B)$

Each party performs one of two dichotomic measurements. The observables are chosen as follows.

(8)$A_1=B_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),A_2=B_2=\left(\begin{array}{cc}0& e^{-i\frac{\pi }{3}}\\ e^{i\frac{\pi }{3}}& 0\end{array}\right)$

The resulting CHSH empirical model is specified by the joint probability distributions shown in Table 1.

In the sheaf-theoretic formulation, the measurement scenario is given by X = {A₁, A₂, B₁, B₂} and O = {0, 1}, with the set of maximal contexts.

(9)$M=\{\{A_1,B_1\},\{A_1,B_2\},\{A_2,B_1\},\{A_2,B_2\}\}$

For each context C ∈ M, the set Ɛ(C) consists of all outcome assignments s:C→O.

(10)$E\left(\right\{A,B\left\}\right)=\{AB\mapsto 00,AB\mapsto 01,AB\mapsto 10,AB\mapsto 11\}$

Each row of Table 1 specifies a probability distribution over Ɛ(C). For example, the first row corresponds to a distribution as follows.

(11)$e_{\{A_1,B_1\}}\in D_{R}E\left(\{A_1,B_1\}\right)$

The resulting family of distributions e = {e_C}_C∈M constitutes a well-defined sheaf-theoretic empirical model. Since it is derived from quantum mechanics, it satisfies the compatibility (no-signalling) conditions, ensuring that the distributions on overlapping contexts agree on their common marginals.

2.2 Characterizing contextuality and nonlocality

2.2.1 Global sections and hidden-variable models

A central insight of the sheaf-theoretic framework is that both contextuality and nonlocality can be characterized in terms of the existence of global sections of a distribution presheaf. In this approach, classicality is identified with the possibility of consistently extending locally observed statistics to a single global probabilistic model defined over all measurements simultaneously. Contextuality and nonlocality arise precisely when such an extension is impossible.

Let X denote the set of all measurements in a given scenario, and let Ɛ be the event sheaf assigning to each subset C ⊆ X the set Ɛ(C) = O^C of outcome assignments for measurements in C. A global section s ∈ Ɛ(X) specifies a definite outcome for every measurement, independently of the context in which it is performed. Such global assignments therefore correspond to classical (deterministic) hidden variable models. A probabilistic hidden-variable model is represented by a distribution d ∈ D_RƐ(X) over global sections. Each deterministic assignment s ∈ Ɛ(X) induces a Dirac distribution δ_s ∈ D_RƐ(X), defined by δ_s(s) = 1 and δ_s(s′) = 0 for s′ ≠ s. The restriction of δ_s to a context C yields the deterministic distribution δ_s|_C ∈ D_RƐ(C). Given a distribution d over global sections, the empirical statistics for each context C are recovered by marginalization.

(12)$e_C\left(s\right)={d|}_C\left(s\right)=\sum _{\begin{array}{c}{s^{\prime }\in E\left(X\right)s^{\prime }|}_C=s\end{array}}d\left(s^{\prime }\right)=\sum _{s^{\prime }\in E\left(X\right)}{\delta }_{{s^{\prime }|}_C}\left(s\right)d\left(s^{\prime }\right)$

This shows explicitly that the observed probabilities arise from averaging over deterministic hidden variables. For each deterministic assignment s and context C, the induced distribution factorizes over individual measurements.

(13)${{\delta }_s|}_C\left(s^{\prime }\right)=\prod _{x\in C}{\delta }_{s\mid \left\{x\right\}}(s^{\prime }\mid \{x\left\}\right)$

This reflects the fact that deterministic hidden variables assign outcomes to each measurement independently of the context in which it appears. These observations lead to the following fundamental equivalence:

Proposition 1. An empirical model admits a global section if and only if it can be realized by a deterministic hidden-variable model.

2.2.2 Linear characterization of global sections

We now reformulate the problem of determining whether an empirical model admits a global section as a linear feasibility question. Let M be a measurement cover of X. Consider the disjoint union of local sections, ${⨿}_{C\in M}E\left(C\right)$, and enumerate its elements as ${\left\{s_i\right\}}_{i=1}^p$. Similarly, enumerate all global sections of Ɛ as ${\left\{t_j\right\}}_{j=1}^q\subseteq E\left(X\right)$. The incidence matrix M is the p × q Boolean matrix defined by the following equation.

(14)$M[i,j]=\{\begin{array}{cc}1,& t_j\mid C=s_i(s_i\in E(C\left)\right)\\ 0,& \text{otherwise}\end{array}$

Each column of M corresponds to a global assignment and records the local sections to which it restricts; each row identifies the set of global assignments compatible with a given local section. Equivalently, M provides a matrix representation of the restriction map.

(15)$E\left(X\right)⟶\prod _{C\in M}E\left(C\right),s\mapsto {\left({s|}_C\right)}_{C\in M}$

This construction depends only on the measurement cover M and the event presheaf Ɛ and is therefore independent of any particular choice of probability theory. It may be applied uniformly to empirical models defined with respect to an arbitrary distribution functor D_R. Given an empirical model {e_C}_C∈M valued in D_R, each local section s_i ∈ Ɛ(C) is assigned a weight e_C(s_i) in the underlying semiring R. Collecting these weights over all contexts yields a vector v ∈ R^p, with components v[i] = e_C(s_i). A putative global distribution is represented by a vector x ∈ R^q, whose entries assign weights to the global sections {t_j}. The requirement that the global distribution reproduces the empirical distributions on every context is encoded by the linear system Mx = v. To ensure normalization, one may augment M with an additional row of ones and append a corresponding entry 1 to v, obtaining an augmented system M′x = v′. The existence of a solution x to this system characterizes whether the empirical model admits a global section. In this way, the question of noncontextuality is recast as a linear feasibility problem, providing a direct and operational criterion for contextuality and nonlocality.

Proposition 2. Solutions of the augmented system M′x = v′ in R are in one-to-one correspondence with global sections realizing the empirical model^[19].

Example 3. We now analyze the empirical model associated with the CHSH scenario introduced in Example 2. In the sheaf-theoretic framework, the existence of a noncontextual or LHV model is equivalent to the existence of a probability distribution on the set of global assignments Ɛ(X) whose marginals reproduce the observed empirical data. Concretely, this amounts to finding a solution to a linear system over the reals, subject to the constraints X_i ≥ 0 and ∑_iX_i = 1, where the variables X_i represent the weights assigned to global outcome assignments. For the CHSH model^[7], the marginal constraints implied by the joint probability distributions in Table 1 lead to the following linear equations.

(16)$X_1+X_2+X_3+X_4=1/2$

(17)$X_2+X_4+X_6+X_8=1/8$

(18)$X_3+X_4+X_{11}+X_{12}=1/8$

(19)$X_1+X_5+X_9+X_{13}=1/8$

Adding the last three equations yields the Equation (20).

(20)$\begin{array}{cc}& X_1+X_2+X_3+X_4+X_5+\\ & X_6+X_8+X_9+X_{11}+X_{12}+X_{13}=3/8\end{array}$

Since all variables X_i are required to be non-negative, the left-hand side of this equation must be greater than or equal to the left-hand side of the first equation. However, the first equation fixes that sum to be 1/2, leading to a contradiction. We therefore conclude that no nonnegative solution exists. This establishes that the CHSH empirical model admits no global probability distribution compatible with all its marginals. In the sheaf-theoretic language, the model has no global section, thereby witnessing Bell nonlocality and the impossibility of a LHV realization.

We now relate global sections to the more familiar notion of hidden-variable models. Let Λ denote a set of hidden variables. A hidden-variable model specifies, for each λ ∈ Λ and each context C ∈ M, a distribution $h_C^{\lambda }\in DE\left(C\right)$, together with a context-independent distribution h_Λ ∈ D(Λ) over hidden variables. For each fixed λ, the family ${\left\{h_C^{\lambda }\right\}}_{C\in M}$ is required to be compatible.

(21)${h_C^{\lambda }|}_{C\cap C^{\prime }}={h_{C^{\prime }}^{\lambda }|}_{C\cap C^{\prime }}$

This corresponds to parameter independence in Bell scenarios. We now consider a general notion of hidden-variable models and analyze the precise relationship between classicality, understood as the factorizability of the empirical model, and the existence of global sections. A hidden-variable model h is said to realize an empirical model e if the empirical probabilities are recovered by averaging over the hidden variables. Explicitly, for all C ∈ M and all s ∈ Ɛ(C).

(22)$e_C\left(s\right)=\sum _{\lambda \in \Lambda }{h}_C^{\lambda }\left(s\right)\cdot h_{\Lambda }\left(\lambda \right)$

The explanatory power of a hidden-variable model lies in its ability to reproduce the statistics of an empirical model, such as quantum mechanics, while attributing them to underlying variables with more classical behaviour. To achieve this, the model must satisfy the property of factorizability, which incorporates both Bell locality and noncontextuality at the level of probability distributions. A hidden-variable model h is said to be factorizable if, for every context C ∈ M and every section s ∈ Ɛ(C).

(23)$h_C^{\lambda }\left(s\right)=\prod _{m\in C}{h}_C^{\lambda }\mid \left\{m\right\}(s\mid \{m\left\}\right)$

This condition states that the probability assigned to a joint outcome factors into probabilities assigned to individual measurement outcomes. If m ∈ C ∩ C′, the compatibility condition implies $h_C^{\lambda }\left|\right\{m\}=h_{C^{\prime }}^{\lambda }|\left\{m\right\}$, ensuring that the outcome distribution for a single measurement does not depend on the context in which it is measured.

Proposition 3. Let e be an empirical model defined on a measurement cover M for a distribution functor D_R. The following are equivalent^[19]:

1. e has a realization by a factorizable hidden-variable model.

2. e has a global section.

More intuitively, factorizability means that, conditioned on any value of the hidden variable, the probability assigned to a joint outcome decomposes into a product of probabilities for individual outcomes. Within the sheaf-theoretic framework, this notion admits a precise and unifying formulation. An empirical model {e_C}_C∈M is said to be extendable to a global section if there exists a distribution d ∈ D_RƐ(X) such that d|C = e_C, for all C ∈ M. The existence of such a global section provides a necessary and sufficient condition for classicality. In Bell scenarios, extendability to a global section is equivalent to Bell locality^[6], while in general measurement scenarios it coincides with noncontextuality in the sense of the KS theorem^[3]. From this viewpoint, nonlocality appears as a special instance of contextuality arising from a particular structure of the measurement cover. The sheaf-theoretic framework therefore yields a single, theory-independent criterion for classical correlations: an empirical model is classical if and only if it admits a global section. Contextuality and nonlocality arise precisely when such a section does not exist. This unifying perspective has led to a broad range of structural and conceptual insights^{[19,55,59-62]}, and serves as a cornerstone for modern approaches to the study of contextuality and nonlocality.

2.2.3 Logical and strong contextuality

While the existence of a global section provides a probabilistic criterion for classicality, the sheaf-theoretic framework also admits a finer classification based solely on the support structure of an empirical model. This possibilistic perspective ignores the precise numerical values of probabilities and retains only the distinction between possible and impossible events.

For an empirical model e = {e_C}_C∈M, let supp(e_C) denote the support of the distribution associated with context C.

(24)$\mathrm{supp}\left(e_C\right)=\{s\in E(C)\mid e_C(s)\ne 0\}$

A global assignment s ∈ Ɛ(X) is said to be consistent with the support of e if the following holds.

(25)${s|}_C\in \mathrm{supp}\left(e_C\right)\forall C\in M$

The set of all such globally compatible assignments is denoted by Equation (26).

(26)$S_e\left(X\right)=\{s\in E(X)\mid s{|}_C\in \mathrm{supp}(e_C)\forall C\in M\}$

The set S_e(X) contains all candidate deterministic hidden-variable assignments that are compatible with the locally possible events specified by the empirical model. If a global distribution d ∈ D_RƐ(X) realizes the empirical model, then every assignment in the support of d must belong to S_e(X). Consequently, S_e(X) provides a canonical possibilistic candidate for a global explanation of the empirical data.

At the possibilistic level, one asks whether the locally possible events admit a globally consistent Booleanvalued description. A possibilistic extension exists if every locally possible section arises as the restriction of some globally compatible assignment. Since every possibilistic extension must be supported on S_e(X), the set S_e(X) may be regarded as a canonical candidate for such an extension. However, the existence of globally compatible assignments is a weaker requirement than the existence of a possibilistic extension. In particular, it may happen that some assignments in the supports of the local distributions cannot be obtained as restrictions of any element of S_e(X). This leads naturally to a hierarchy of contextuality based on the support structure of empirical models.

Definition 4. An empirical model e is said to be logically contextual if there exists a context C ∈ M and a section s ∈ supp(e_C) such that no element of S_e(X) restricts to s on C.

An even stronger obstruction arises when no globally compatible assignment exists at all.

Definition 5. An empirical model e is said to be strongly contextual if the following holds.

(27)$S_e\left(X\right)=\varnothing$

Strong contextuality represents the strongest possibilistic form of contextuality. In this case, there is no assignment of outcomes to all measurements that remains compatible with every locally possible event. Thus, not only does the model fail to admit a global probabilistic realization, it fails even at the level of possibilistic consistency.

Within the Abramsky-Brandenburger framework, contextuality admits a natural hierarchy according to the strength of the obstruction to global consistency. Strong contextuality implies logical contextuality, which in turn implies probabilistic contextuality, while the converse implications generally fail. These three levels correspond respectively to the nonexistence of any globally compatible assignment, the failure of certain locally possible events to extend globally, and the impossibility of reproducing the empirical probabilities by a global distribution. The hierarchy is illustrated schematically in Figure 4. Representative examples occur at each level. The CHSH empirical model^[7] is probabilistically contextual but not logically contextual, since its support admits a possibilistic extension even though no global probability distribution reproduces the observed correlations. Hardy’s nonlocality model^[21] provides a canonical example of logical contextuality without strong contextuality, as some locally possible events admit no globally compatible extension despite the existence of globally consistent assignments. Strong contextuality is exhibited by the GHZ-Mermin model^[63,64], the Peres-Mermin magic square^[24,36], the 18-vector KS model^[22], and Popescu-Rohrlich correlations^[57], for which no globally compatible assignment exists at all. This hierarchy reveals that contextuality is not a binary property but admits different degrees of obstruction to classical explanations.

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Figure 4. The Abramsky-Brandenburger hierarchy of contextuality. Adapted from reference^[19]. CC BY 4.0. Strong contextuality $\left(S_e\right(X)=\varnothing )$ excludes the existence of any globally compatible assignment. Logical contextuality occurs when some locally possible events admit no globally compatible extension, despite $S_e\left(X\right)\ne \varnothing$. Probabilistic contextuality arises when a possibilistic extension exists but no global probability distribution reproduces the empirical model. The implications are strict: Strong Contextuality ⊊ Logical Contextuality ⊊ Probabilistic Contextuality. Typical examples are the Peres-Mermin square^[24,36], Hardy nonlocality^[21], and the CHSH model^[7]. CHSH: Clauser-Horne-Shimony-Holt.

2.3 Unified perspective and outlook

The sheaf-theoretic framework provides a unified and explicitly theory-independent characterization of contextuality and nonlocality, formulated entirely in terms of the global consistency structure of empirical models^[19,55,65]. At its core lies a single organizing principle: an empirical model is classically realizable if and only if its locally compatible marginals admit a global section. From this perspective, contextuality and nonlocality arise precisely as obstructions to global extendability, rather than as phenomena tied to any particular mathematical formalism of quantum theory. This viewpoint recasts the role of hidden-variable theories and measurement incompatibility at a fundamentally structural level. Factorizable hidden-variable models correspond exactly to empirical models that admit global sections, while the failure of such models reflects not the absence of underlying variables, but the impossibility of constructing a jointly consistent assignment compatible with all observable marginals. In this sense, classical explanations fail not because probability theory itself breaks down, but because the local statistical data cannot be glued into a single global probabilistic model. Likewise, measurement incompatibility, often attributed to the noncommutativity of quantum observables, admits a more general interpretation. In the sheaf-theoretic framework, it can be understood as the nonexistence of a joint probability distribution whose marginals reproduce the observed empirical statistics^[66,67]. In this sense, incompatibility reflects a structural obstruction imposed by global consistency requirements on probability assignments, rather than a specifically quantum postulate.

Beyond its unifying explanatory role, the sheaf-theoretic framework naturally delineates several promising directions for further development. One important direction is the extension from predominantly finite scenarios to fully measure-theoretic settings, enabling the systematic treatment of continuous-variable and infinite dimensional systems within the same formalism^[68]. A second major challenge concerns multipartite and high-dimensional scenarios, where the combinatorial and algebraic complexity of measurement covers grows rapidly^[56,69-71]. Addressing these regimes may require new categorical, algebraic, or cohomological tools capable of classifying and quantifying obstructions to global consistency in a scalable manner^[50,72,73]. In this context, a deeper comparison with topos-theoretic approaches to quantum theory appears particularly promising. While both frameworks interpret contextuality in terms of the failure of global sections, clarifying the precise relationship between operationally defined measurement covers, empirical distributions, and quantum-specific structures such as the spectral presheaf may help distinguish genuinely universal features of quantum correlations from those tied to specific theoretical formalisms. Finally, the framework provides a natural setting for analyzing quasi-probability and negative-probability representations of quantum phenomena^[74,75]. Historically, such representations, ranging from the Wigner function to related phase-space constructions, have signaled the impossibility of realizing a classical joint distribution compatible with all observable marginals, rather than any internal inconsistency of probability theory itself^[51-54]. Interpreted sheaf-theoretically, they suggest the possibility of generalized empirical models in which global sections exist only in a quasi-probabilistic sense. Exploring the operational and conceptual significance of such generalized models may further clarify the precise limits of classical explanations of quantum correlations.

In summary, the sheaf-theoretic framework provides a rigorous and unified characterization of contextuality and nonlocality in terms of global consistency and its obstruction. Far from being a mere technical reformulation of known results, it supplies a flexible and conceptually transparent language for understanding the deep structural relations between contextuality and nonlocality, and offers a promising foundation for future investigations across a wide range of physical theories.

3. Graph-Theoretic Approach

While the sheaf-theoretic framework provides a high-level and mathematically powerful approach to unify contextuality and nonlocality, it remains largely abstract with respect to the operational constraints, and the connection to experimentally measurable quantities is often nontrivial. In particular, the sheaf-based description does not explicitly represent how individual measurement events exclude one another, nor how these exclusivity relations restrict the strength of attainable correlations. These limitations motivate alternative approaches that retain conceptual rigor while offering greater operational transparency. The graph-theoretic approach provides a complementary perspective by encoding measurement events as vertices and representing pairwise exclusivity by edges, thereby offering a compact and operationally transparent description of the underlying constraint structure. This combinatorial perspective enables a direct comparison between classical, quantum, and more general probabilistic models at the level of the event structure itself. In this section, we provide an overview of this framework systematically, beginning with the construction of exclusivity graphs from operational scenarios and proceeding to the interpretation of the corresponding graph-theoretic constraints.

3.1 The graph-theoretic framework

There are two main approaches underlying graph-theoretic formulations of quantum correlations: the compatibility-hypergraph approach and the exclusivity graph approach. In the compatibility-hypergraph approach, the primary focus is on the compatibility relations among a finite set of measurements, with correlations arising from the structure of jointly measurable contexts; comprehensive accounts of this framework can be found in the studies^[18,76,77]. By contrast, the exclusivity-graph approach characterizes correlations in terms of exclusivity relations among a finite set of measurement events. Within this framework, the convex sets of correlations obtained in classical, quantum, and more general probabilistic theories satisfying the Exclusivity Principle are naturally related to fundamental convex sets in graph theory. This review will focus on the latter perspective.

To formalize correlations in an experimentally meaningful way, we adopt a general operational framework. Two types of interventions are assumed to be available: preparations and operations. A fundamental requirement is reproducibility: each preparation and each operation can be implemented arbitrarily many times, allowing outcome probabilities to be estimated from relative frequencies. For a given preparation, an operation may yield several possible outcomes, each occurring with a well-defined probability. Correlations are described in terms of probabilities assigned to operationally defined events. Two preparations are said to be operationally equivalent if they give rise to identical outcome probabilities for all possible operations; each such equivalence class defines a state. The resulting state space is assumed to be convex, with extremal points corresponding to pure states. Operations with multiple outcomes are referred to as measurements, and two measurements are equivalent if they generate identical outcome statistics for all states. A complete mathematical description of the system therefore consists of a set of states, a set of measurements, and a rule assigning outcome probabilities to each measurement-state pair. Such a description constitutes a probabilistic model, and a probability theory is understood as a collection of such models.

We restrict attention to outcome-repeatable measurements, whose outcomes are stable under sequential application to the same physical system. A measurement j is outcome-repeatable if, whenever outcome k is obtained, any subsequent measurement of j yields the same outcome with unit probability. Let P(k|j) denote the probability of outcome k upon performing measurement j, with the dependence on the preparation left implicit. In general probabilistic theories, a set of measurements need not admit a joint probability distribution. When such a distribution exists for all states, the measurements are said to be compatible.

An event is defined as an equivalence class of outcome-measurement pairs that are assigned the same probability for all states. Thus, outcome a of measurement x and outcomes b, …, c of measurements y, …, z are equivalent if P(a|x) = P(b, …, c|y, …, z) for every state. Two events are exclusive if they correspond to compatible measurements and cannot occur simultaneously. The exclusivity relations of a correlation experiment are encoded in an exclusivity graph G, whose vertices represent events and whose edges connect exclusive pairs. This representation encodes the combinatorial structure of events underlying a correlation experiment and provides a unified framework for characterizing classical, quantum, and more general correlations through graph-theoretic quality.

Example 4. As an illustration of this framework, we reconsider the standard CHSH scenario^[7], introduced previously in Example 2, with the explicit purpose of constructing its associated exclusivity graph. In this scenario, there are four dichotomic measurements labeled 0, 1, 2, 3, each with outcomes 0 and 1. Measurements 0 and 2 are performed by Alice, while measurements 1 and 3 are performed by Bob. Joint measurements are performed only for the four compatible measurement pairs (0, 1), (1, 2), (2, 3), and (3, 0) on the same prepared states. Each joint outcome (a, b|x, y) constitutes a measurement event and will be represented as a vertex in the graph.

The exclusivity relation between events is defined operationally: two events are exclusive if they cannot occur simultaneously in a single run of the experiment. In the CHSH scenario, such exclusivity arises whenever two events assign different outcomes to at least one joint measurement. Edges in the graph are introduced to encode this exclusivity relation. Applying these rules yields the full exclusivity graph of the CHSH scenario, denoted by G_CHSH. This graph contains 16 vertices, corresponding to all joint events associated with the compatible measurement settings, with edges connecting all pairs of mutually exclusive events. Maximal sets of pairwise exclusive events form cliques in G_CHSH, that is, each fixed joint measurement setting (x, y) gives rise to a clique of size four, corresponding to the four possible joint outcomes.

In Figure 5, vertices represent events and edges represent exclusivity relations. For graphical clarity only, sets of pairwise exclusive events are arranged along straight lines or circumferences. The CHSH expression S_CHSH involves only a subset of the measurement events. Restricting G_CHSH to the eight events appearing in the inequality, and retaining all exclusivity relations among them, defines an induced subgraph, denoted by G_CHSH. By construction, such a subgraph is obtained by selecting all the blue vertices with edges between them present in the original graph. The resulting graph G_CHSH is isomorphic to the eight-vertex circulant graph Ci₈(1, 4).

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Figure 5. The exclusivity graph of the CHSH scenario. Reproduced with permission from reference^[7]. Copyright © 1969 American Physical Society. CHSH: Clauser-Horne-Shimony-Holt.

The correlations can be expressed as positive linear combinations of event probabilities^[78]:

(28)$S=\sum _i{w}_{i}P\left(e_i\right)$

where w_i > 0 are weights and P(e_i) denotes the probability of event e_i. The corresponding inequalities are as follows.

(29)$S_{\mathrm{CHSH}}=\sum _{i=0}^3\sum _{a,b}P(a,b\mid i,i+1)\stackrel{\mathrm{LHV}}{\le }3$

The sum over outcomes a, b∈{0, 1} obeys a = b for i ≠ 2 and a ≠ b for i = 2, and the index i + 1 is taken modulo 4. The labels “LHV” indicate the bounds predicted by local hidden variable models^[6,7,79].

While the example above uses uniform weights w_i = 1, general expressions may assign different weights to different events. Any such expression S can be encoded as a vertex-weighted exclusivity graph (G, w):

• Each vertex i ∈ V(G) represents an event e_i contributing to S;

• Edges denote pairs of exclusive events;

• The weight function $w:V\to R_{+}$ assigns the coefficient w_i to each event.

This formalism enables a combinatorial characterization of contextuality and nonlocality, where physical limits on S can be associated with the graph parameters. The exclusivity graph thereby serves as a powerful tool for classifying and analyzing quantum correlations.

3.2 Combinatorial bounds on physical theories

Once a correlation experiment is encoded in terms of an exclusivity graph G a natural and fundamental question arises: what ultimately constrains the strength of the correlations that can be assigned to the vertices of G? Remarkably, a broad class of physical theories, including classical, quantum, and general probabilistic models, can be characterized in terms of the combinatorial structure of G, together with a basic probabilistic consistency requirement known as the exclusivity principle^[80,81]. The exclusivity principle, originally introduced by Specker, asserts that the probabilities associated with any set of pairwise exclusive events must sum to at most 1^[82,83]. In both classical and quantum theories this constraint is automatically satisfied, as it follows from the existence of an underlying probability measure or a quantum state, respectively. However, for general probabilistic models no such guarantee exists, and violations of the exclusivity principle can occur^[84]. Throughout this section, we denote by E1 the class of theories that satisfy the exclusivity principle when it is imposed only at the level of a single exclusivity graph G^[27,81].

One of the central results of the graph-theoretic framework is that the maximum value of a correlation expression S attainable within different classes of physical theories is completely characterized by specific graph invariants^[27].

(30)$S\stackrel{\mathrm{LHV},\mathrm{NCHV}}{\le }\alpha (G,w)\stackrel{Q}{\le }\vartheta (G,w)\stackrel{E1}{\le }{\alpha }^{\ast }(G,w)$

Here α(G, w) denotes the independence number of the weighted graph (G, w)^[85], corresponding to classical models admitting deterministic and noncontextual value assignments^[3]. The quantity $\vartheta (G,w)$ is the Lovász number^[86,87], which characterizes the maximum value achievable within quantum theory^[18]. Finally, α*(G, w) is the fractional packing number^[88], describing the maximal value allowed by theories satisfying the exclusivity principle E1, according to which the probabilities associated with the vertices of any clique of G must sum to at most 1^[27].

In classical hidden-variable models, both local in Bell scenarios and noncontextual in single-system settings, measurement outcomes are assumed to be predetermined. Consequently, each event is assigned a definite value 0 or 1, and probabilities arise as convex mixtures of such deterministic assignments. Since pairwise exclusive events cannot both be assigned value 1, the set of events simultaneously assigned value 1 must form an independent (or stable) set of the graph G. The maximal classical value of S is therefore given by the maximum total weight of an independent set, which is the weighted independence number α(G, w)^[85].

In quantum mechanics, exclusivity between events is implemented by the orthogonality of the corresponding projective measurements. To formalize this structure, we adopt the standard graph-theoretic notion of an orthonormal representation (OR). An orthonormal representation of a graph G in $R^d$ assigns a nonzero unit vector $|v_i⟩\in R^d$ to each vertex i ∈ V(G) such that $⟨v_i\mid v_j⟩=0$, whenever i and j are nonadjacent in G. In the framework of exclusivity graphs, it is convenient to work with the complement $\overline{G}$ of a graph G, which is the graph with vertex set V such that two vertices i, j are adjacent in $\overline{G}$ if and only if i, j are not adjacent in G. Given an orthonormal representation of $\overline{G}$ and a normalized state vector $|\psi ⟩$, referred to as the handle, one considers the quantity $\sum _{i\in V\left(G\right)}{w}_i{\left|⟨\psi \mid v_i⟩\right|}^2$. The maximum of this expression, taken over all choices of the handle $|\psi ⟩$, is the Lovász number $\vartheta (G,w)$. From an operational perspective, this construction captures the quantum realization of events as projection operators Π_i acting on a Hilbert space of suitable dimension, with probabilities given by $p_i=⟨\psi |{\Pi }_i|\psi ⟩$. For rank-one projectors, one may identify $|v_i⟩={\Pi }_i|\psi ⟩/\sqrt{⟨\psi |{\Pi }_i|\psi ⟩}$, which yields an orthonormal representation of $\overline{G}$ with handle $|\psi ⟩$. Consequently, the maximal value of the correlation expression S attainable within quantum theory coincides with the Lovász number $\vartheta (G,w)$^[86].

For general probabilistic theories constrained only by the exclusivity principle E1, the maximal value of the correlation expression S is given by the fractional packing number α*(G, w) := max∑_i∈V(G)w_ip_i, where the maximization is taken over all assignments p_i ≥ 0 satisfying the clique constraints ∑_i∈Cp_i ≤ 1 for every clique C ⊆ G. Each clique corresponds to a set of pairwise exclusive events, and these linear constraints implement the exclusivity principle directly at the probabilistic level. The quantity α*(G, w) therefore characterizes the strongest correlations compatible with the exclusivity structure encoded by G under this minimal consistency requirement.

The three graph parameters appearing in Equation (30) differ markedly in their computational complexity. Determining the independence number α(G, w) is NP-hard in general, even in the unweighted case^[89]. By contrast, the Lovász number $\vartheta (G,w)$ can be computed to arbitrary precision in polynomial time using semidefinite programming^[85]. The fractional packing number α*(G, w) is defined by a linear program with clique constraints. While the optimization problem itself is efficiently solvable once the relevant clique inequalities are specified, generating the full list of cliques of an arbitrary graph is NP-hard in general^[85].

Example 5. The unifying power of the graph-theoretic framework becomes transparent when applied to paradigmatic correlation inequalities. In the CHSH scenario^[7], the associated exclusivity graph yields the bounds α = 3, $\vartheta =2+\sqrt{2}$, and α* = 4, corresponding respectively to the maximal values attainable within LHV theories^[7], quantum theory^[90], and no-signaling models^[57]. In the KCBS scenario^[44], one similarly finds α = 2, $\vartheta =\sqrt{5}$, and α* = 5/2, reproducing the established bounds for noncontextual hidden-variable theories^[44], quantum theory^[78], and generalized no-disturbance models^[58]. Despite their distinct operational and physical interpretations, both scenarios are captured by the same combinatorial hierarchy encoded in Equation (30).

The Lovász number $\vartheta (G,w)$ provides a universal upper bound on the quantum value of any Bell or noncontextual inequality associated with a vertex-weighted exclusivity graph (G, w). A fundamental result of the graph-theoretic approach shows that, for every graph G, there always exists a noncontextuality inequality whose maximum quantum violation is exactly given by $\vartheta \left(G\right)$. Moreover, in such scenarios, the set of quantum-achievable probability assignments to the vertices of G coincides with the Grötschel-Lovász-Schrijver theta body TH(G)^[85]. However, this correspondence does not always hold in Bell scenarios. Not every graph G admits a Bell inequality that achieves the quantum bound $\vartheta \left(G\right)$, due to additional constraints imposed by Bell scenarios, which enforce a specific labeling of measurement events^[91]. This provides further evidence that nonlocality constitutes a special subclass of contextuality. To some extent, the notion of contextuality abstracts away from specific experimental realizations and thereby significantly simplifies the characterization of the quantum set, compared with approaches that rely on a fixed labeling of measurement events^[69,90,92].

In analogy with the definition of the quantum set Q(G), one can introduce the corresponding sets for classical theories and for more general probabilistic theories satisfying the exclusivity principle E1. We denote by C(G) the convex hull of all deterministic noncontextual assignments:

(31)$C\left(G\right):=\mathrm{conv}\{x^S:S\text{is a stable set of}G\}$

where a deterministic assignment x^S ∈ {0, 1}^|V| is associated with a stable set S ⊆ V(G) via $x_i^S=1$ if i ∈ S and $x_i^S=0$ otherwise. Similarly, we define Ɛ¹(G) as the polytope of all probability assignments satisfying the exclusivity principle E1.

(32)$E^1\left(G\right):=\{p\in R_{+}^{\left|V\right|}:\sum _{i\in C}{p}_i\le 1\text{for all cliques}C\}$

With these definitions, one obtains the inclusion as follows.

(33)$C\left(G\right)\subseteq Q\left(G\right)\subseteq E^1\left(G\right)$

Both C(G) and Ɛ¹(G) are polytopes, whereas Q(G) is generally not. A natural question is under which conditions distinguish quantum from classical correlations, and under which conditions the exclusivity principle E1 singles out the quantum set. It is proven that C(G) = Q(G) and Q(G) = Ɛ¹(G) hold if and only if the graph G contains no induced odd cycle C_n with n ≥ 5, nor the complement $\overline{C_n}$ of such a cycle^[27,85,93]. Remarkably, the answer is entirely graph-theoretic and is characterized by the absence of certain induced subgraphs.

3.3 Summary and further perspectives

The graph-theoretic approach provides a conceptually transparent and operationally grounded framework for analyzing quantum correlations. By taking measurement events as fundamental and encoding their mutual exclusivity in a graph, one abstracts away from specific experimental setups while retaining the essential constraints that any probabilistic theory must satisfy. In this sense, the exclusivity graph is not merely a mathematical convenience, but the natural language for expressing the structure of correlations and the limits imposed by classical, quantum, or more general probabilistic theories.

A central insight of this framework is that the maximal correlations attainable under different theories are determined by intrinsic combinatorial properties of the graph. Classical correlations are bounded by the independence structure of the graph, reflecting the limitations of predetermined, noncontextual assignments. Quantum correlations surpass classical bounds, with orthogonality imposing geometric consistency, and the Lovász number provides as a precise quantification of this quantum advantage. More general probabilistic theories are constrained only by the exclusivity principle, which enforces logical consistency among mutually exclusive events. In this way, the framework connects operational, geometric, and combinatorial features.

It is important to emphasize that the inclusion of nonlocality within the framework of contextuality should not be interpreted as rendering nonlocality redundant. While modern approaches such as the sheaf-theoretic and graph-theoretic frameworks identify nonlocality as a particular manifestation of contextuality, this structural relationship does not diminish the independent significance of Bell nonlocality. In particular, nonlocality is intrinsically associated with spatially separated systems, relativistic causality, entanglement distribution, and device-independent information processing. Consequently, describing nonlocality as a special case of contextuality should be understood as a statement about the mathematical and structural organization of quantum correlations, rather than as a claim that contextuality can fully replace nonlocality as a foundational concept. Although contextuality and nonlocality are often discussed as originating from the same quantum resource, nonlocality retains its unique significance in spatially separated quantum systems and device-independent quantum information protocols.

Beyond these formal results, the graph-theoretic perspective highlights the conceptual unity of contextuality and nonlocality. Both phenomena arise from the impossibility of assigning globally consistent truth values to all events, with quantum mechanics allowing correlations that surpass classical bounds while remaining limited by the exclusivity structure encoded in the graph. This perspective unifies previously distinct notions, emphasizing that the distinctive features of quantum correlations are fundamentally about the combinatorial structure of mutually exclusive events.

Looking forward, the graph-theoretic approach highlights several open questions and promising directions. In multipartite scenarios, additional causal, spatial, or labeling constraints may further restrict the set of achievable quantum correlations, emphasizing the connections between the graph-theoretic principle and experimental structure. More broadly, one may ask whether principles based solely on exclusivity, perhaps supplemented by consistency requirements under graph composition, are sufficient to uniquely characterize the quantum set within the broader landscape of no-disturbance or no-signaling models. Thus, the graph-theoretic framework not only provides a powerful tool for designing and classifying correlation experiments, but also offers a unified perspective on the logical, physical, and structural essence of quantum contextuality and nonlocality.

4. Experimental Aspects

The conceptual unification of contextuality and nonlocality, as formalized within the sheaf-theoretic and graph-theoretic framework, provides a powerful abstraction for characterizing nonclassical correlations independently of Hilbert space representations. While this theoretical framework captures the logical and topological structure of quantum correlations, experimental realization is crucial for validating these insights and for exploring their practical consequences. The experimental challenge lies in translating the abstract notions of global sections, measurement compatibility, and exclusivity into physically implementable protocols, often constrained by measurement sharpness, detection efficiency, and noise. In the remainder of this section, we review several important recent photonics experiments that investigate the connections between contextuality and nonlocality.

4.1 Revealing nonlocality from local contextuality

A particularly instructive class of experiments demonstrates how Bell nonlocality can be operationally revealed from single-system contextuality through sequential compatible measurements. Following a proposal by Cabello^[94], Liu et al.^[95] showed that Bell nonlocality can be revealed by combining locally contextual correlations with bipartite correlations that, taken alone, remain compatible with LHV models. In this sense, nonlocality is not directly exhibited by standard Bell inequality violations, but is instead activated through the presence of state-independent contextuality (SI-C) in one of the subsystems.

The experiment involves four qubits distributed between two distant parties. Alice holds qubits 1 and 2, while Bob holds qubits 3 and 4. In each experimental run, Alice performs a sequence of three jointly compatible measurements on her subsystem, whereas Bob performs a single measurement on his subsystem, as illustrated in Figure 6. Space-like separation between Alice’s and Bob’s laboratories ensures that no communication can influence their respective measurement choices or outcomes. Experimentally, this configuration is realized using two hyperentangled photons, with spatial and polarization degrees of freedom encoding independent qubits.

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Figure 6. Conceptual illustration of the experiment of Liu et al., demonstrating Bell nonlocality from local contextuality. Reproduced with permission from reference^[95]. Copyright © 2016 American Physical Society. A distributed four-qubit entangled state is shared between two spatially separated parties, Alice and Bob. On Alice’s side, sequential compatible measurements associated with the Peres-Mermin magic square provide a state-independent contextuality witness ⟨χ⟩, the experimentally measured value ⟨χ⟩^Exp exceeds the NCHV bound 4 and therefore reveals contextuality. Correlations between Alice’s and Bob’s observables define a Bell-type quantity ⟨S⟩, which by itself does not violate the corresponding LHV bound and therefore does not reveal Bell nonlocality. However, when the contextuality witness and correlation terms are combined into the hybrid Bell expression ⟨ω⟩ = ⟨χ⟩ + ⟨S⟩. The experimentally measured value ⟨ω⟩^Exp exceeds the local-hidden-variable bound ⟨ω⟩_LHV ≤ 16, thereby revealing Bell nonlocality. NCHV: noncontextual hidden-variable; LHV: local hidden-variable.

Alice’s sequential measurements are chosen from the Peres-Mermin set^[24,35], which exhibits SI-C. Six sequences of three compatible observables generate local triple correlations, while bipartite correlations arise from comparing Alice’s later measurements in each sequence with Bob’s single measurement. The central quantity tested experimentally is the inequality introduced in the study^[94].

(34)$⟨\omega ⟩\equiv ⟨\chi ⟩+⟨S⟩\stackrel{\mathrm{LHV}}{\le }16$

This must be satisfied by all LHV theories^[6,79]. The term ⟨χ⟩ consists exclusively of correlations among Alice’s sequential measurements and corresponds to a Peres-Mermin contextuality inequality χ ≤ 4^[24,35], while ⟨S⟩ contains only bipartite correlations between Alice’s later measurements in each sequence and Bob’s single measurement^[94]. Experimentally, a strong violation of the noncontextual bound is observed in the local term ⟨χ⟩, while the bipartite correlations ⟨S⟩ alone remain compatible with LHV models. Only when both contributions are jointly considered does the inequality ⟨ω⟩ ≤ 16 become violated, thereby revealing Bell nonlocality^[95]. This feature distinguishes the present scheme from standard Bell tests. Here we focus on the conceptual framework and physical intuition underlying the protocol. Detailed discussions of the state preparation, path-polarization encoding, and measurement implementation are beyond the scope of this review and can be found in Ref^[95].

From a conceptual perspective, this experiment does not introduce a new form of nonlocality. Rather, it provides an explicit operational mechanism by which Bell-type violations can emerge from locally contextual measurement structures when distributed across subsystems via entanglement. Although such implementations rely on assumptions of compatibility and non-disturbance inherent to sequential measurements, they establish a clear and experimentally accessible bridge between quantum contextuality and Bell nonlocality. From a sheaf-theoretic perspective, Alice’s sequential compatible measurements define a contextual empirical model that admits no consistent global assignment of outcomes, independently of the quantum state. The entangled distant system does not introduce additional contextual structure, but rather redistributes this incompatibility into a bipartite measurement scenario, where it becomes accessible to experimental tests of Bell-type constraints. In graph-theoretic terms, the Peres-Mermin measurements implemented by Alice correspond to an exclusivity structure whose quantum correlations exceed the limits imposed by noncontextual models. Cabello’s inequality (34) can then be understood as a constraint on a composite exclusivity structure, obtained by embedding this local contextual structure into a larger bipartite scenario through perfect Alice-Bob correlations. The observed violation reflects the impossibility of a single classical probability assignment over the combined set of measurement events. Within the unified sheaf- and graph-theoretic frameworks discussed above, these experiments exemplify how contextuality can be regarded as a more primitive form of nonclassicality, whose structural incompatibility with classical probability theory may be redistributed across subsystems and ultimately manifested as Bell nonlocality under suitable operational conditions.

It is worth emphasizing that the experiment should not be interpreted as a universal criterion for witnessing Bell nonlocality from arbitrary forms of contextuality. In this case, the construction relies on a specific SI-C structure based on the Peres-Mermin set and on a specially designed Bell inequality that combines local contextual correlations with bipartite correlations. While it provides a compelling operational link between contextuality and nonlocality, it does not imply that every contextual scenario can be transformed into a Bell nonlocality test. More general contextuality-to-nonlocality conversion schemes have subsequently been developed for SI-C sets^[96], whereas no analogous universal framework is currently known for arbitrary SD-C scenarios.

4.2 Coexistence of contextuality and nonlocality

Contextuality is commonly divided into SI-C and SD-C, according to whether its observation depends on the prepared quantum state. In an SI-C scenario, a fixed measurement configuration rules out a noncontextual description for every state in the relevant Hilbert space, including the maximally mixed state. State independence should therefore be understood relative to the specified Hilbert space and measurement set: the contextuality is a property of the measurement structure rather than of a specially prepared state. Representative examples are provided by KS sets^[3], for which no context-independent assignment of outcomes is compatible with all measurement contexts. By contrast, SD-C can be revealed only for suitable quantum states. Its observation depends jointly on the measurement configuration and the state under examination. The KCBS inequality for a qutrit is a canonical example^[44]. Although SI-C and SD-C are operationally distinguished by their state dependence, this distinction alone does not establish whether they constitute fundamentally different physical resources or different manifestations of a common contextual structure.

The possibility that nonlocality and contextuality might manifest within a single physical scenario was first systematically analyzed by Kurzyński et al.^[97]. In their proposal, two spatially separated parties perform a standard CHSH Bell test^[7], while one party simultaneously probes SD-C through a KCBS inequality^[44] on a local subsystem, using the same set of compatible measurements. Within quantum theory, as well as in more general no-disturbance theories, this scenario was shown to exhibit a trade-off relation between the corresponding nonlocality and contextuality witnesses, implying that at most one of the two inequalities can be violated^[97]. This result motivated the conjecture of a fundamental monogamy relation between Bell nonlocality and SD-C. The conjectured monogamy was subsequently confirmed experimentally^[98]. Importantly, this type of monogamy constraint does not generally apply to SI-C, whose violation is independent of the quantum state being prepared. Indeed, simultaneous violations of Bell inequalities and state-independent KS inequalities were later reported^[99]. Moreover, the original monogamy argument implicitly relied on the standard Bell scenario in which each party performs a single measurement per experimental run. As pointed out in the study^[100], each measurement appearing in a CHSH inequality can be in principle supplemented with an additional compatible observable, leading to the formulation of more general Bell inequalities incorporating measurement contexts.

Building on this observation, Xue et al. provided both a theoretical construction and a photonic realization in which Bell nonlocality and SD-C are synchronously certified^[101]. The measurement structure is illustrated in Figure 7. Alice chooses between two dichotomic observables, A₁ and A₂, while Bob has five dichotomic observables, B_1, …, B₅, arranged in a KCBS compatibility cycle. In the generalized CHSH test, B₁ constitutes one of Bob’s effective settings, whereas the product B₂B₃ of the outcomes of the compatible observables B₂ and B₃ constitutes the other. The resulting Bell expression is:

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Figure 7. Measurement structure for the synchronous test of Bell nonlocality and state-dependent contextuality. Reproduced with permission from reference^[101]. Copyright © 2023 American Physical Society. A bipartite state ρ_AB is distributed between Alice and Bob. Alice chooses between two dichotomic observables, A₁ and A₂, whereas Bob has five dichotomic observables, B₁, …, B₅, whose compatibility relations form a pentagon, as indicated by the gray dashed edges. Context 1 comprises the joint measurements {A_i, B₁}_i=1,2, while Context 2 comprises {A_i, B₂, B₃}_i=1,2. Since B₂ and B₃ are compatible, their product B₂B₃ constitutes an effective dichotomic observable in the generalized CHSH inequality. Context 3 consists of the five adjacent compatible pairs {B_j, B_j+1}, with mod 5, used to construct the KCBS inequality. The generalized CHSH and KCBS expressions have local-hidden-variable and noncontextual-hidden-variable bounds of 2 and 3, respectively. Their simultaneous violation for the same state and measurement scenario demonstrates the coexistence of Bell nonlocality and state-dependent contextuality. KCBS: Klyachko-Can-Binicioğlu-Shumovsky; CHSH: Clauser-Horne-Shimony-Holt.

(35)$\begin{array}{cc}⟨\mathrm{CHSH}⟩& =⟨A_1{B}_1⟩+⟨A_1{B}_2{B}_3⟩\\ & +⟨A_2{B}_1⟩-⟨A_2{B}_2{B}_3⟩\le 2\end{array}$

where the bound holds for LHV models. The same measurement architecture also provides the adjacent-pair correlations required for the KCBS expression:

(36)$\begin{array}{cc}⟨\mathrm{KCBS}⟩& =⟨B_1{B}_2⟩+⟨B_2{B}_3⟩+⟨B_3{B}_4⟩\\ & +⟨B_4{B}_5⟩-⟨B_5{B}_1⟩\le 3\end{array}$

where the bound applies to noncontextual hidden-variable models. For suitable bipartite quantum states, both inequalities can be violated, thereby demonstrating the coexistence of Bell nonlocality and SD-C within the same generalized measurement scenario.

Here, “synchronous” does not mean that all incompatible measurement settings are applied to the same individual copy of the state. As in other Bell and contextuality experiments, the different contexts are sampled over repeated trials using identically prepared states. The term instead indicates that the two violations are obtained from a single operational scenario, a common state preparation, and a unified measurement architecture.

The earlier monogamy result and the later coexistence experiment are therefore not contradictory. They concern different operational uses of the available data. If the outcomes of additional compatible measurements are discarded when constructing the Bell witness, a trade-off between the CHSH and KCBS violations can arise. If the full compatible context is retained and incorporated into a generalized Bell inequality, the two violations can coexist. The relation between contextuality and nonlocality is consequently not determined by the quantum state alone, but depends crucially on the measurement scenario, its compatibility structure, and which observable data are admitted into the corresponding witnesses. From the unified sheaf- and graph-theoretic perspectives, both phenomena can be represented within a common structural language, while remaining operationally distinct forms of nonclassicality.

4.3 Converting contextuality into nonlocality

Before discussing the contextuality-to-nonlocality conversion scheme, it is important to emphasize that the existing construction applies specifically to SI-C. In SI-C scenarios, contextuality is determined solely by the compatibility structure of the measurement set and can be demonstrated for every quantum state in the underlying Hilbert space. This state-independent character enables a systematic graph-theoretic embedding of the contextuality scenario into a bipartite Bell experiment, as proposed by Cabello, where the nonclassicality originates from the measurement structure itself rather than from a particular state preparation. By contrast, SD-C arises only for the specific prepared states and the chosen measurement scenario. Although Bell nonlocality and SD-C may coexist within suitably designed experimental scenarios, no general procedure is currently known that converts an arbitrary SD-C scenario into an equivalent Bell nonlocality scenario in the same systematic manner as the SI-C construction. Consequently, the contextuality-to-nonlocality conversion discussed below should be understood as a result established for SI-C sets, rather than a universal correspondence applicable to all forms of contextuality.

A critical step toward a fully operational unification of contextuality and nonlocality was proposed by Cabello^[96], who showed that any SI-C set can be systematically embedded into a bipartite Bell scenario. In this construction, one party measures the projectors defining the SI-C set, while the other party measures their complex conjugates. The resulting Bell inequalities are violated by quantum correlations whenever the underlying measurement set is state-independent contextual, thereby allowing contextuality to be manifested through spacelike separated measurements. Conceptually, this approach converts a single-system logical obstruction into a bipartite manifestation of correlations, while bypassing the long-standing “compatibility” and “sharpness” loopholes^{[45,46,102-105]} inherent to sequential-measurement tests of contextuality^[47,48].

From an experimental perspective, Bell tests derived from SI-C sets must also confront well-known loopholes that arise in realistic implementations^[106]. In particular, the detection loophole and the associated fair sampling assumption have historically played a central role in photonic Bell experiments. Early optical tests often relied on the assumption that detected events faithfully represent the entire ensemble of emitted photon pairs, leaving open the possibility that undetected events could bias the observed correlations. A particularly influential treatment of this problem was developed by Eberhard^[107], who introduced Bell inequalities that explicitly incorporate detection and no-detection outcomes. By enlarging the outcome alphabet to include “no-click” events, the Eberhard formulation allows Bell violations to be demonstrated under significantly lower detection efficiencies while remaining robust against background counts. Within the conceptual framework emphasized in this review, such “click/no-click” patterns can be interpreted as deterministic hidden-variable strategies defined over an enlarged outcome space. Despite these challenges, high-dimensional photonic orbital angular momentum (OAM) platforms^[108] provide a natural route toward implementing Cabello’s conversion scheme^[96]. OAM modes can be used to define an infinitely dimensional discrete Hilbert space, and the number of effective dimensions can be readily tailored as required^[109,110]. Moreover, spontaneous parametric down-conversion naturally produces photon pairs entangled in OAM^[111], enabling the preparation of high-dimensional entangled states suitable for Bell tests constructed from SI-C sets. Spatially separated measurements allow the required projective measurements to be implemented without invoking sequential compatibility assumptions.

An experimental implementation of Cabello’s conversion scheme was reported by Sheng et al.^[106] (Figure 8), using high-dimensional OAM-entangled photon pairs to test Bell inequalities derived from representative SI-C sets. These include the simplest SI-C set in dimension d = 3 identified by Yu and Oh^[112], the minimal KS set in dimension d = 4 proposed by Cabello et al.^[22], and the KS set with the smallest number of contexts in dimension d = 6 introduced by Lisoněk et al.^[113]. By combining entanglement concentration techniques with optimized holographic measurement strategies, these experiments achieved statistically significant violations of the corresponding Bell inequalities across multiple dimensions. Within this construction, the observed violations probe the contextual structure encoded by the SI-C measurement sets, while the result should not be interpreted as a general conversion applicable to all forms of contextuality.

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Figure 8. Scheme of the experiment by Sheng et al. Reproduced with permission from reference^[106]. Copyright © 2025 American Physical Society. The inset (a) shows the original two-photon OAM spectrum of limited spiral bandwidth before entanglement concentration, while (b) shows the maximally entangled OAM spectrum after concentration. OAM: orbital angular momentum; BBO: beta barium borate; BS: beam splitter; SLM: spatial light modulator; BPF: bandpass filter; SMF: single-mode fiber; SPCM: single-photon counting module.

From a unified theoretical perspective, Cabello’s conversion scheme admits a transparent interpretation within both the sheaf-theoretic and graph-theoretic frameworks discussed above. In sheaf-theoretic terms, a SI-C set defines an empirical model that admits no global section independently of the quantum state. The bipartite embedding redistributes this obstruction across spatially separated measurements, rendering it testable as a Bell-type constraint. In graph-theoretic language, the orthogonality relations of the SI-C set define an exclusivity graph whose noncontextual bound is given by its independence number; the associated Bell inequality inherits this bound, while quantum correlations attain values governed by the corresponding Lovász number. Experimental implementations then realize these abstract structures by engineering entangled states and measurement architectures that faithfully reproduce the required exclusivity relations. The experiments demonstrate that contextuality can function as a more primitive quantum resource from which nonlocality can be operationally derived. Unlike earlier approaches that reveal nonlocality through auxiliary contextual correlations or demonstrate their coexistence under generalized measurement scenarios, Cabello’s framework provides a direct and loophole-resistant route for certifying contextuality via genuine Bell tests. In this sense, high-dimensional photonic platforms, particularly those based on OAM entanglement, emerge as a conceptually clean and experimentally powerful arena for realizing the deepest structural connections between contextuality and nonlocality.

Taken together, the experiments reviewed in this section demonstrate how the abstract unification of contextuality and nonlocality articulated within the sheaf-theoretic and graph-theoretic frameworks can be progressively translated into concrete laboratory protocols. The sheaf-based perspective identifies contextuality and nonlocality as manifestations of a common obstruction to global assignments, while graph-theoretic formulations encode these obstructions into experimentally testable inequalities defined by exclusivity relations. Experimental implementations then realize these abstract structures by engineering specific measurement architectures and quantum states that faithfully reproduce the underlying compatibility and exclusivity patterns. Early experiments established operational bridges in which locally contextual structures give rise to Bell-type violations under suitable correlation analyses, while subsequent coexistence experiments demonstrated that contextuality and nonlocality can be simultaneously observed within a single composite system. More recently, conversion-based schemes grounded in SI-C have shown that contextual correlations can be certified through genuinely bipartite Bell tests, thereby bypassing key loopholes associated with sequential measurements. In this way, experimental progress has followed a clear conceptual trajectory: from logical and topological characterizations, through combinatorial witnesses, to operational protocols that render the unified structure of quantum correlations empirically accessible.

5. Conclusions

In this review, we have surveyed and synthesized recent developments that connect contextuality and nonlocality across sheaf-theoretic, graph-theoretic, and experimental frameworks. The sheaf-theoretic framework provides a rigorous, representation-independent characterization of contextuality and nonlocality, highlighting the logical and topological obstructions to classical explanations through the language of global sections and presheaves. This high-level perspective reveals the intrinsic incompatibilities of quantum events, offering a unifying conceptual lens for understanding nonclassical correlations.

Building on this foundation, the graph-theoretic framework translates the abstract structural insights into combinatorial and operationally meaningful terms. By representing measurement events as vertices and their exclusivity relations as edges, this framework captures the bounds of classical, quantum, and generalized probabilistic theories bounds through graph invariants such as the independence number, the Lovász number, and the fractional packing number. In doing so, it bridges formal theory and experimental practice, providing systematic prescriptions for constructing noncontextuality and Bell inequalities, as well as quantitative predictions for their quantum violations.

Sheaf-theoretic and graph-theoretic approaches offer complementary perspectives on contextuality and nonlocality. The sheaf framework encodes measurement scenarios as presheaves, with local sections representing compatible event assignments and the existence of a global section signaling noncontextuality. Graph-theoretic methods, in contrast, map events to vertices and exclusivity relations to edges, with independent sets and graph invariants quantifying classical, quantum, and generalized bounds. These two formalisms are closely connected: the set of local sections in the sheaf approach corresponds to independent sets in the graph, and the Lovász theta number provides a concrete numerical reflection of the cohomological obstructions that indicate contextuality in the sheaf framework. While sheaf theory excels in generality and formal rigor, the graph approach translates abstract structural constraints into operationally accessible and computationally tractable quantities, facilitating experimental design and numerical analysis. Together, they provide a unified understanding, with sheaf theory clarifying the underlying conceptual structure and graph theory enabling practical quantification.

Recent experimental advances, with a particular focus on photonic implementations, have demonstrated the practical relevance of this unified viewpoint, from sequential measurements on single photons to high-dimensional entanglement experiments. These implementations validate the operational content of both sheaf- and graph-based descriptions and, crucially, reveal explicit pathways for relating contextuality and nonlocality. In particular, the embedding of SI-C sets into bipartite Bell scenarios exemplifies how abstract unification directly informs experimental design, enabling robust and scalable tests of high-dimensional quantum correlations.

In summary, the developments reviewed in this article illustrate how sheaf-theoretic, graph-theoretic, and experimental approaches provide complementary perspectives on the relationship between contextuality and nonlocality. Taken together, these approaches clarify how different manifestations of quantum nonclassicality can be analyzed within common structural frameworks while retaining important operational distinctions. They also highlight several open questions concerning contextuality-to-nonlocality conversion, resource-theoretic characterizations, and applications in high-dimensional quantum information processing.

6. Outlooks

Despite the substantial progress reviewed in this article, several important challenges remain open. A central theoretical question concerns the precise physical origin of the connection between contextuality and nonlocality. Although modern sheaf-theoretic and graph-theoretic frameworks provide a unified mathematical description in which Bell nonlocality appears as a particular form of contextuality, it remains unclear whether both phenomena can ultimately be derived from a common underlying physical principle. In particular, while SI-C can be systematically embedded into Bell scenarios, no general conversion framework is currently known for arbitrary SD-C scenarios. Determining whether such a framework exists, and identifying the conditions under which contextuality can be transformed into nonlocality, remain important open problems.

Another promising direction is the development of unified resource-theoretic approaches. Contextuality and nonlocality are both recognized as valuable resources for quantum information processing, yet their quantitative relationship and possible resource interconversion mechanisms remain poorly understood. Establishing a common resource framework could provide a deeper operational understanding of quantum correlations and clarify how different forms of nonclassicality contribute to quantum advantages.

From an experimental perspective, high-dimensional photonic platforms, particularly those based on structured light and OAM, offer unique opportunities for investigating contextuality and nonlocality within a common physical system. Future experiments may explore more sophisticated contextuality–nonlocality coexistence scenarios, test contextuality-to-nonlocality conversion schemes in increasingly higher dimensions, and investigate the role of contextuality in quantum-enhanced protocols. However, significant challenges remain, including the verification of compatibility and nondisturbance conditions, the closure of relevant experimental loopholes, and the realization of scalable, high-fidelity measurements in large Hilbert spaces.

Beyond their foundational significance, contextuality and nonlocality are increasingly recognized as key resources for future quantum technologies. Bell nonlocality has long served as the cornerstone of device-independent quantum communication, randomness certification, and quantum key distribution, whereas contextuality has recently emerged as an essential resource for quantum computation, quantum advantage, and magic-state-based fault-tolerant architectures. Understanding whether these resources can be described within a common operational framework may reveal deeper connections between seemingly distinct quantum information tasks and shed new light on the origin of quantum advantages. Importantly, the identification of Bell nonlocality as a special case of contextuality should be understood as a statement about mathematical structure rather than physical redundancy. Nonlocality retains its distinctive significance through its connection with spacelike separation, relativistic causality, and device-independent quantum information processing. Recent studies further suggest that contextuality and nonlocality may constitute complementary resources, with contextuality primarily associated with local quantum advantages and nonlocality governing distributed quantum information processing. In particular, the development of large-scale quantum networks raises new questions regarding the interplay between contextuality and nonlocality. Future distributed quantum architectures may involve both local contextual resources and nonlocal entanglement-assisted correlations operating simultaneously across different network layers. Determining how these resources can be combined, transformed, and optimized for communication, computation, and sensing tasks remains largely unexplored. High-dimensional photonic systems based on structured light and OAM provide a particularly promising platform for addressing these questions, as their large Hilbert-space capacity enables the simultaneous exploration of contextuality, nonlocality, and quantum communication protocols within a common experimental architecture. Progress on these questions may contribute to a more unified operational understanding of nonclassical correlations and clarify how local and distributed quantum advantages are related within a common conceptual framework.

Authors contribution

Chen L: Conceptualization, writing-review & editing, supervision.

Sheng J: Conceptualization, investigation, writing-original draft.

Zhang D: Writing-review & editing.

Conflicts of interest

Lixiang Chen is an Associate Editor of Light Manipulation and Applications. The other authors declare no conflicts of interest.

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Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 12034016 and 12205107), the National Key R&D Program of China (Grant No. 2023YFA1407200), Natural Science Foundation of Fujian Province of China (Grant No. 2021J02002) for Distinguished Young Scientists (Grant No. 2015J06002), Program for New Century Excellent Talents in University (Grant No. NCET-13-0495), Natural Science Foundation of Xiamen City (Grant No. 3502Z20227033), and Fundamental Research Funds for the Central Universities (Grant No. ZQN-1206).

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© The Author(s) 2026.