  Triple-photon states generated by three-mode spontaneous parametric
down-conversion are the paradigm of unconditional non-Gaussian states,
essential assets for quantum advantage. How to fully characterize their
non-Gaussian entanglement remains however elusive. We propose here a hierarchy
of sufficient and necessary conditions for separability of the broad family of
spontaneously-generated three-mode non-Gaussian states. We further derive
state-of-the-art conditions for genuine tripartite non-Gaussian entanglement,
the strongest class of entanglement. We apply our criteria to triple-photon
states revealing that they are fully inseparable and genuinely entangled in
moments of order 3n. Our results establish a systematic framework for
characterizing the entanglement of triple-photon states and thus fostering
their application in quantum information protocols.
