  We derive a theoretical framework for the experimental certification of
non-Gaussian features of quantum states using double homodyne detection. We
rank experimental non-Gaussian states according to the recently defined stellar
hierarchy and we propose practical Wigner negativity witnesses. We simulate
various use-cases ranging from fidelity estimation to witnessing Wigner
negativity. Moreover, we extend results on the robustness of the stellar
hierarchy of non-Gaussian states. Our results illustrate the usefulness of
double homodyne detection as a practical measurement scheme for retrieving
information about continuous variable quantum states.
