  We study the faces of the set of quantum correlations, i.e., the Bell and
noncontextuality inequalities without any quantum violation. First, we
investigate the question whether every proper (tight) Bell inequality for two
parties, other than the trivial ones from positivity, normalization and
no-signaling can be violated by quantum correlations, i.e., whether the
classical Bell polytope or the smaller correlation polytope share any facets
with their respective quantum sets. To do this, we develop a recently derived
bound on the quantum value of linear games based on the norms of game matrices
to give a simple sufficient condition to identify linear games with no quantum
advantage. Additionally we show how this bound can be extended to the general
class of unique games, illustrating it for the case of three outcomes. We then
show as a main result that the paradigmatic examples of correlation Bell
inequalities with no quantum violation, namely the non-local computation games
do not constitute tight Bell inequalities, not even for the correlation
polytope. We also extend this to an arbitrary prime number of outcomes for a
specific class of these games. We then study the faces in the simplest CHSH
Bell scenario of binary dichotomic measurements, and identify edges in the set
of quantum correlations in this scenario.
  Finally, we relate the non-contextual polytope of single-party correlation
inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the
compatibility graph of observables in the contextuality scenario and $\nabla G$
denotes the suspension graph of $G$. We observe that there exist tight
non-contextuality inequalities with no quantum violation, and furthermore that
this set of inequalities is beyond those implied by the Consistent Exclusivity
principle.
