We establish a strict logical separation between two distinct phenomena in orthogonality hypergraphs: chromatic completeness, the possibility of assigning a single globally consistent nondegenerate spectrum to all contexts, and geometric coordinatizability, the existence of a faithful orthogonal representation by rays. A strong chromatic number larger than the Hilbert-space dimension obstructs only the former. It does not, by itself, obstruct the existence of a faithful orthogonal representation. We make this separation explicit by comparing two three-dimensional examples with the same strong chromatic number. A completed 25-ray version of the Yu-Oh configuration has strong chromatic number four and nevertheless possesses an explicit faithful orthogonal representation in R^3. Conversely, Greechie's G_{32} hypergraph also has strong chromatic number four, and has a separating and unital set of two-valued states, but we give an elementary algebraic proof that it admits no faithful orthogonal representation in C^3. The obstruction in G_{32} is therefore not chromatic but projective-geometric: the incidence relations force two distinct atoms to collapse onto the same ray.
We establish a strict logical separation between two distinct phenomena in orthogonality hypergraphs: chromatic completeness, the possibility of assigning a single globally consistent nondegenerate spectrum to all contexts, and geometric coordinatizability, the existence of a faithful orthogonal representation by rays. A strong chromatic number larger than the Hilbert-space dimension obstructs only the former. It does not, by itself, obstruct the existence of a faithful orthogonal representation. We make this separation explicit by comparing two three-dimensional examples with the same strong chromatic number. A completed 25-ray version of the Yu-Oh configuration has strong chromatic number four and nevertheless possesses an explicit faithful orthogonal representation in R^3. Conversely, Greechie's G_{32} hypergraph also has strong chromatic number four, and has a separating and unital set of two-valued states, but we give an elementary algebraic proof that it admits no faithful orthogonal representation in C^3. The obstruction in G_{32} is therefore not chromatic but projective-geometric: the incidence relations force two distinct atoms to collapse onto the same ray.
Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a classical realization with n-uniform outcomes per context. Consequently, it cannot represent a “completable” noncontextual set of coexisting n-ary outcomes per maximal observable. This result serves as a chromatic analogue of the Kochen-Specker theorem. We present an explicit example of a four-colorable quantum logic in dimension three. Furthermore, chromatic contextuality suggests a novel restriction on classical truth values, thereby excluding two-valued measures that cannot be extended to n-ary colorings. Using this framework, we establish new bounds for the house, pentagon, and pentagram hypergraphs, refining previous constraints.
Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for n∈{2,4,8}, any graph G which admits a real n-dimensional orthogonal representation is quantumly n-colorable. This result can be recast as the statement that the real sphere Sn−1 is quantumly n-colorable for these values of n. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide candidate finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if Sn−1 is quantumly n-colorable, then either n=2 or n is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order n exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the existence of a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that Sn−1 admits a rank-one quantum n-coloring if and only if n∈{2,4,8}, thereby settling a conjecture of Zeng and Zhang, as does the fact that for all m≥1, there exists a catalytic zero-error remote state preparation protocol for real m-qubit states with m bits of communication and which consumes m ebits.
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.
This paper argues that the quantum measurement problem is not a single explanatory gap but a collection of distinct tasks. We separate detector representability, probability structure, single-event actuality, and outcome-conditioned refinement, and distinguish all four from causal registration, the physical criterion by which an already-instantiated event becomes a completed measurement record. These are explanatory roles within one measurement phenomenon rather than successive temporal stages. The analysis begins from the contrast between a global quantum description containing several detector-correlated components and the single entry that appears in a realized measurement history. Rather than compressing this contrast into an unexplained transition often labeled “collapse,” we ask which specific explanatory constraints must be satisfied before the relation between the two descriptions can be diagnosed precisely. A finite-resolution alternating position–momentum history is used throughout to separate questions of record formation, probability, actuality, causal registration, and outcome-conditioned refinement. A continuum analysis shows that making the registered history arbitrarily dense does not by itself recover a classical trajectory. Several parts of the architecture are exact once their inputs are specified. A detector representation induces an exact quotient; a specified acquisition channel and record family induce system-side effects; and a specified quantum instrument determines a family of subnormalized conditional branches before any one outcome is instantiated. Given a realized outcome label, the same instrument determines the corresponding normalized outcome-conditioned state. A finite-dimensional acquisition model explicitly realizes the representability chain, while a record-algebra indistinguishability result shows that removing coherence between detector-record sectors can leave all expectation values in the lifted record algebra unchanged even when that coherence remains detectable in a richer context. These results sharpen, but do not dissolve, the measurement problem. Record structure does not derive Born weights; a probability law does not itself constitute the event realized in one run; a mathematically available conditional branch family does not select one branch condition as actual; and causal onset, stabilization, registration, and conditional refinement all presuppose or follow that event rather than produce it. The resulting constraint audit tests—without assuming—the hypothesis that global quantum multiplicity and detector-record definiteness may belong to different representational levels. The main conclusion is not a new collapse dynamics but a localization of the remaining explanatory burden. Once representability, probability, causal registration, and conditional refinement are assigned their proper roles, the unresolved actuality transition can no longer be hidden inside the single label “collapse.” In the typed measurement architecture considered here, no pre-event operation has the mathematical role of producing an individual event token without additional event-producing structure. This is version 1.0 of the manuscript.
Observable-based Kochen-Specker proofs are configurations of multi-qubit Pauli observables grouped into contexts whose products are plus or minus the indentity. Their robustness as state-independent contextuality tests can be measured by the tolerated error per context ε=2d/∣H∣, where d is the contextuality degree and ∣H∣ the number of contexts. Since the degree depends only on an underlying abstract structure called the hypergram i.e. the pair formed by the context hypergraph and the anticommutation graph, the search for highly contextual proofs can be carried out on them instead, with no reference to qubits or to any particular Pauli labeling. We further exploit this by enumerating anticommutation graphs first, and then by associating to each graph G the single hypergram carrying its entire hypergraph support HS(G), so that exactly one candidate is examined per graph. Applied to the House of Graphs database and to censuses of vertex-transitive graphs on at most 24 vertices, this pipeline recovers the Peres-Mermin square, the doily and the Mermin pentagram, and yields configurations reaching ε=0.707, against 0.424 for the previous published record. The best configurations are predominantly those stemming from line graphs and unions of graphs; we explain the former by showing that every perfect matching of a graph is a context of its line graph, which exhibits the Peres-Mermin square and the doily as the first members of two infinite families. We close with finite geometric descriptions of the most striking configurations inside symplectic polar spaces, in terms of ovoids, hyperbolic quadrics and Fano planes.
We discuss representations and colorings of orthogonality hypergraphs in terms of their two-valued states interpretable as classical truth assignments. Such hypergraphs, if they allow for a faithful orthogonal representation, have quantum mechanical realizations in terms of intertwined contexts or maximal observables that are widely discussed as empirically testable criteria for contextuality. A reconstruction is possible for the class of perfectly separable hypergraphs. Colorings can be constructed from a minimal set of two-valued states. Some examples from exempt categories are presented, which either cannot be reconstructed by two-valued states or whose two-valued states cannot yield a chromatic number that is equal to the maximal clique number.