  We present an invertible map between correlations in any bipartite Bell
scenario and behaviours in a family of contextuality scenarios. The map takes
local, quantum and non-signalling correlations to non-contextual, quantum and
contextual behaviours, respectively. Consequently, we find that the membership
problem of the set of quantum contextual behaviours is undecidable, the set
cannot be fully realised via finite dimensional quantum systems and is not
closed. Finally, we show that neither this set nor its closure is the limit of
a sequence of computable supersets, due to the result MIP*=RE.
