  Photon coincidences represent an important resource for quantum technologies.
They expose nonlinear quantum processes in matter and are essential for sources
of entanglement. We derive broadly applicable criteria for quantum non-Gaussian
two-photon coincidences that certify a new quality of photon sources. The
criteria reject states emerging from Gaussian parametric processes, which often
limit applications in quantum technologies. We also analyse the robustness of
the quantum non-Gaussian coincidences and compare with the heralded quantum
non-Gaussianity of single-photons based on them.
