Quantum violation of stochastic noncontextual hidden-variable theories

We formulate stochastic noncontextual hidden-variable theories. In such theories the hidden state {λ} specifies not definite values, but expectation values A{}({λ}), . . . of observables A, . . . and noncontextuality means that A{}({λ}) is independent of which other commuting observables commuting with A are measured together with A. We show via Bell inequalities that such theories conflict with quantum theory and propose a two-photon experimental test. We also show that for a single particle of spin (2ⁿ-1)/2, quantum violation of classical noncontextuality grows exponentially with n.