  We derive a family of inequalities involving different phase-space
distributions of a quantum state which have to be fulfilled by any classical
state. The violation of these inequalities is a clear signature of
nonclassicality. Our approach combines the characterization of nonclassical
effects via negativities in phase-space distributions with inequality
conditions usually being formulated for moments of physical observables.
Importantly, the obtained criteria certify nonclassicality even when the
involved phase-space distributions are non-negative. Moreover, we show how
these inequalities are related to correlation measurements. The strength of the
derived conditions is demonstrated by different examples, including squeezed
states, lossy single-photon states, and even coherent states.
