  We present two results on the subject of quantum contextuality and
cohomology, and non-locality and quantum advantage with shallow circuits.
Abramsky et al. showed that a range of examples of quantum contextuality is
detected by a cohomological invariant based on \v{C}ech cohomology. However,
the approach does not give a complete cohomological characterisation of
contextuality. A different cohomological approach to contextuality was
introduced by Okay et al. Their approach exploits the algebraic structure of
the Pauli operators and their qudit generalisations known as Weyl operators. We
give an abstract account of this structure, then generalise their approach to
any example of contextuality with this structure. We prove at this general
level that the approach does not give a more complete characterisation of
contextuality than the \v{C}ech cohomology approach.
  Bravyi, Gosset, and K\"{o}nig (BGK) gave the first unconditional proof that a
restricted class of quantum circuits is more powerful than its classical
analogue. The result, for the class of circuits of bounded depth and fan-in
(shallow circuits), exploits a particular family of examples of contextuality.
BGK's quantum circuit and computational problem are derived from a family of
non-local games related to the well-known GHZ non-local game. We present a
generalised version of their construction. A systematic way of taking examples
of contextuality and producing unconditional quantum advantage results with
shallow circuits.
