  We consider a range of "theories" that violate the uncertainty relation for
anti-commuting observables derived in [JMP, 49, 062105 (2008)]. We first show
that Tsirelson's bound for the CHSH inequality can be derived from this
uncertainty relation, and that relaxing this relation allows for non-local
correlations that are stronger than what can be obtained in quantum mechanics.
We continue to construct a hierarchy of related non-signaling theories, and
show that on one hand they admit superstrong random access encodings and
exponential savings for a particular communication problem, while on the other
hand it becomes much harder in these theories to learn a state. We show that
the existence of these effects stems from the absence of certain constraints on
the expectation values of commuting measurements from our non-signaling
theories that are present in quantum theory.
