We present new inequalities for testing Leggett's model [Found. Phys. 33, 1469 (2003)] of non-local quantum correlations. Remarkably these inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions. The simplest of these inequalities is experimentally violated. We also investigate more general models a la Leggett, and show that such models must have vanishing marginals in order to reproduce the correlations of the singlet state.
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