  We devise a new class of criteria to certify the nonclassicality of photon-
and phonon-number statistics. Our criteria extend and strengthen the broadly
used Klyshko's criteria, which require knowledge of only a finite set of
Fock-state probabilities. This makes the criteria well-suited to experimental
implementation in realistic conditions. Moreover, we prove the completeness of
our method in some scenarios, showing that, when only two or three Fock-state
probabilities are known, it detects all finite distributions incompatible with
classical states. In particular, we show that our criteria detect a broad class
of noisy Fock states as nonclassical, even when Klyshko's do not. The method is
directly applicable to trapped-ion, superconducting circuits, and optical and
optomechanical experiments with photon-number resolving detectors. This work
represents a significant milestone towards a complete characterisation of the
nonclassicality accessible from limited knowledge of the Fock-state
probabilities.

Quantum Physics