  One of the central foundational questions of physics is to identify what
makes a system quantum as opposed to classical. One seminal notion of
classicality of a quantum system is the existence of a non-contextual hidden
variable model as introduced in the early work by Bell, Kochen and Specker. In
quantum optics, the non-negativity of the Wigner function is a ubiquitous
notion of classicality. In this work we establish an equivalence between these
two concepts. In particular, we show that any non-contextual hidden variable
model for Gaussian quantum optics has an alternative non-negative Wigner
function description. Conversely, it was known that the Wigner representation
provides a non-negative non-contextual description of Gaussian quantum optics.
It follows that contextuality and Wigner negativity are equivalent notions of
non-classicality and equivalent resources for this quantum subtheory. In
particular, both contextuality and Wigner negativity are necessary for a
computational speed-up of quantum Gaussian optics. At the technical level, our
result holds true for any subfamily of Gaussian measurements that include
homodyne measurements, i.e., measurements of standard quadrature observables.
