  In quantum optics, nonclassicality of quantum states is commonly associated
with negativities of phase-space quasiprobability distributions. We argue that
the impossibility of any classical simulations with phase-space functions is a
necessary and sufficient condition of nonclassicality. The problem of such
phase-space classical simulations for particular measurement schemes is
analysed in the framework of Einstein-Podolsky-Rosen-Bell's principles of
physical reality. The dual form of this problem results in an analogue of Bell
inequalities. Their violations imply the impossibility of phase-space classical
simulations and, as a consequence, nonclassicality of quantum states. We apply
this technique to emblematic optical measurements such as photocounting,
including the cases of realistic photon-number resolution and homodyne
detection in unbalanced, balanced, and eight-port configurations.
