We propose a hierachy of nonclassicality criteria in phase space. Our formalism covers the negativity in phase space as a special case and further adresses nonclassicality for quantum states with positive phase-space distributions. Remarkably, it enables us to detect every nonclassical Gaussian state and every finite dimensional state in Fock basis by looking into only three phase-space points. Furthermore, our approach provides an experimentally accessible lower bound for the nonclassicality measure based on trace distance. We also extend our method to detecting genuine quantum non-Gaussianity of a state with a non-negative Wigner function. We finally establish our formalism by employing generalized quasiprobability distributions to demonstrate its power for a practical test using an on-off detector array.
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