The Kochen–Specker (KS) theorem is a cornerstone result in quantum foundations, establishing that quantum correlations in Hilbert spaces of dimension d ≥ 3 cannot be explained by (consistent) hidden variable theories that assign a single deterministic outcome to each measurement. Specifically, there exist finite sets of vectors in these dimensions such that no non-contextual deterministic ({0, 1}) outcome assignment is possible obeying the rules of exclusivity and completeness—that the sum of assignments to every set of mutually orthogonal vectors be ≤1 and the sum of value assignments to any d mutually orthogonal vectors be equal to 1. Another central result in quantum foundations is Gleason’s theorem that justifies the Born rule as a mathematical consequence of the quantum formalism. The KS theorem can be seen as a consequence of Gleason’s theorem and the logical compactness theorem. In a similar vein, Gleason’s theorem also indicates the existence of KS-type finite vector constructions to rule out other finite-alphabet outcome assignments beyond the {0, 1} case. Here, we propose a generalisation of the KS theorem that rules out hidden variable theories with outcome assignments in the set {0, p, 1 − p, 1} for p ∈ [0, 1/d) ∪ (1/d, 1/2]. The case p = 1/2 is especially physically significant. We show that in this case the result rules out (consistent) hidden variable theories that are fundamentally binary, i.e., theories where each measurement has fundamentally at most two outcomes (in contrast to the single deterministic outcome per measurement ruled out by KS). We present a device-independent application of this generalised KS theorem by constructing a two-player non-local game for which a perfect quantum winning strategy exists (a Pseudo-telepathy game) while no perfect classical strategy exists even if the players are provided with additional no-signaling resources of PR-box type (with marginals in {0, 1/2, 1}).

Kochen-Specker （KS） 定理是量子基础的基础结果，它确定了维度 d ≥ 3 的希尔伯特空间中的量子相关性不能用（一致的）隐藏变量理论来解释，这些理论为每个测量分配了一个确定性结果。具体来说，这些维度中存在有限的向量集，因此在遵守排他性和完备性规则的情况下，不可能进行非上下文确定性 （{0， 1}） 结果分配——每组互交向量的赋值之和为 ≤1，任何 d 个互交向量的值赋值之和等于 1。量子基础的另一个核心结果是格里森定理，该定理证明了波恩规则是量子形式主义的数学结果。KS 定理可以看作是格里森定理和逻辑紧缩定理的结果。同样，格里森定理也表明存在 KS 型有限向量结构，以排除 {0， 1} 情况以外的其他有限字母结果分配。在这里，我们提出了 KS 定理的推广，排除了结果分配在集合 {0， p， 1 − p， 1} 中 p∈ [0， 1/d） ∪ （1/d， 1/2] 的隐藏变量理论。情况 p = 1/2 在物理上特别重要。我们表明，在这种情况下，结果排除了（一致的）隐藏变量理论，这些理论基本上是二元的，即每个测量基本上最多有两个结果的理论（与 KS 排除的每个测量的单一确定性结果相反）。 我们通过构建一个双人非局部博弈来提出这个广义 KS 定理的独立于设备的应用，该博弈存在完美的量子获胜策略（伪心灵感应博弈），而即使为玩家提供了额外的 PR-box 类型的无信号资源（{0， 1/2， 1} 中的边际），也不存在完美的经典策略。