We present an invertible map between correlations in any bipartite Bell scenario and behaviors in a family of contextuality scenarios. The map takes local, quantum, and no-signaling correlations to noncontextual, quantum, and contextual behaviors, respectively. Consequently, we find that the membership problem of the set of quantum contextual behaviors is undecidable, the set cannot be fully realized 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.