A wide variety of photonic quantum information processing tasks requires non-Gaussian quantum states, from entanglement distillation to universal quantum computing and quantum error-correction. Characterizing and understanding the properties of these states is therefore of major importance. 

In this talk, I will present the stellar representation, which is a recent phase-space formalism allowing to characterise non-Gaussian quantum states in photonic quantum information processing. The main quantity of interest in this context is the so-called stellar rank, which provides a measure of the non-Gaussian character of quantum states. I will explain its operational properties, how it can be used to assess state preparation and state conversion protocols in quantum photonics, how it can be measured experimentally and how it quantifies the computational usefulness of non-Gaussian quantum states.