  The use of quantum information in technology promises to supersede the
so-called classical devices used nowadays. Understanding what features are
inherently non-classical is crucial for reaching better-than-classical
performance. This thesis focuses on two nonclassical behaviours: quantum
contextuality and Wigner negativity. The former is a notion superseding
nonlocality that can be exhibited by quantum systems. To date, it has mostly
been studied in discrete-variable scenarios. In those scenarios, contextuality
has been shown to be necessary and sufficient for advantages in some cases. On
the other hand, negativity of the Wigner function is another unsettling
non-classical feature of quantum states that originates from phase-space
formulation in continuous-variable quantum optics. Continuous-variable
scenarios offer promising candidates for implementing quantum computations.
Wigner negativity is known to be a necessary resource for quantum speedup with
continuous variables. However contextuality has been little understood and
studied in continuous-variable scenarios.
  We first set out a robust framework for properly treating contextuality in
continuous variables. We also quantify contextuality in such scenarios by using
tools from infinite-dimensional optimisation theory. Building upon this, we
show that Wigner negativity is equivalent to contextuality in continuous
variables with respect to Pauli measurements thus establishing a
continuous-variable analogue of a celebrated result by Howard et al. We then
introduce experimentally-friendly witnesses for Wigner negativity of single
mode and multimode quantum states, based on fidelities with Fock states, using
again tools from infinite-dimensional optimisation theory. We further extend
the range of previously known discrete-variable results linking contextuality
and advantage into a new territory of information retrieval.
