  We devise a method to certify nonclassical features via correlations of
phase-space distributions by unifying the notions of quasiprobabilities and
matrices of correlation functions. Our approach complements and extends recent
results that were based on Chebyshev's inequality [Phys. Rev. Lett. 124, 133601
(2020)]. The method developed here correlates arbitrary phase-space functions
at arbitrary points in phase space, including multimode scenarios and
higher-order correlations. Furthermore, our approach provides necessary and
sufficient nonclassicality criteria, applies to phase-space functions beyond
$s$-parametrized ones, and is accessible in experiments. To demonstrate the
power of our technique, the quantum characteristics of discrete- and
continuous-variable, single- and multimode, as well as pure and mixed states
are certified only employing second-order correlations and Husimi functions,
which always resemble a classical probability distribution. Moreover, nonlinear
generalizations of our approach are studied. Therefore, a versatile and broadly
applicable framework is devised to uncover quantum properties in terms of
matrices of phase-space distributions.
