  Quantum non-Gaussian states of photons and phonons are conclusive and direct
witnesses of higher-than-quadratic nonlinearities in optical and mechanical
processes. Moreover, they are proven resources for quantum sensing,
communication and error correction with diverse continuous-variable systems.
This review introduces theoretical analyses of nonclassical and quantum
non-Gaussian states of photons and phonons. It recapitulates approaches used to
derive operational criteria for photons tolerant to optical losses, their
application in experiments and their nowadays extension to quantum non-Gaussian
photon coincidences. It extends to a recent comparison of quantum
non-Gaussianity, including robustness to thermal noise, and sensing capability
for high-quality phononic Fock states of single trapped cooled ions. The review
can stimulate further development in the criteria of quantum non-Gaussian
states and experimental effort to prepare and detect such useful features,
navigating the community to advanced quantum physics and technology.
