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Minimal Kochen-Specker theorem in finite dimensions

In [1] we proved a strengthened Kochen-Specker theorem in 3 dimensions: non-contextual hidden variable (NCHV) models cannot reproduce all the quantum correlations of two compatible observables, which is a minimal requirement imposed on the NCHV models. Here we shall exclude the NCHV models with this minimal requirement in $d\ge 4$ dimensions by state-independent and experimentally testable inequalities satisfied by all NCHV models that are required to reproduce only the quantum correlations of at most two compatible observables. Furthermore our proofs use the smallest number of rays known so far, e.g., 25 (instead of 31) rays in $5$ dimensions and $5d-2\lfloor d/3\rfloor$ rays in $d\ge 6$ dimensions.