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Quantum theory predicts and experiments confirm that nature can producecorrelations between distant events that are nonlocal in the sense of violatinga Bell inequality. Nevertheless, Bell's strong sentence {\it Correlations cryout for explanations} remains relevant. The maturing of quantum informationscience and the discovery of the power of nonlocal correlations, e.g. forcryptographic key distribution beyond the standard Quantum Key Distributionschemes, strengthen Bell's wish and make it even more timely.  In 2003, Leggett proposed an alternative model for nonlocal correlations[Found. Phys. 33, 1469 (2003)], that he proved to be incompatible with quantumpredictions. We present here a new approach to this model, along with newinequalities for testing it. Remarkably these inequalities can be derived in avery simple way, assuming only the non-negativity of probability distributions;they are also stronger than previously published Leggett-type inequalities. Thesimplest of these inequalities is experimentally violated. Then we go beyondLeggett's model, and show that one cannot ascribe even partially definedindividual properties to the components of a maximally entangled pair.

Testing quantum correlations versus single-particle properties within Leggett's model and beyond