  It has long been known that the existence of certain superquantum nonlocal
correlations would cause communication complexity to collapse. The absurdity of
a world in which any nonlocal binary function could be evaluated with a
constant amount of communication in turn provides a tantalizing way to
distinguish quantum mechanics from incorrect theories of physics; the statement
"communication complexity is nontrivial" has even been conjectured to be a
concise information-theoretic axiom for characterizing quantum mechanics. We
directly address the viability of that perspective with two results. First, we
exhibit a nonlocal game such that communication complexity collapses in any
physical theory whose maximal winning probability exceeds the quantum value.
Second, we consider the venerable CHSH game that initiated this line of
inquiry. In that case, the quantum value is about 0.85 but it is known that a
winning probability of approximately 0.91 would collapse communication
complexity. We provide evidence that the 0.91 result is the best possible using
a large class of proof strategies, suggesting that the communication complexity
axiom is insufficient for characterizing CHSH correlations. Both results build
on new insights about reliable classical computation. The first exploits our
formalization of an equivalence between amplification and reliable computation,
while the second follows from an upper bound on the threshold for reliable
computation with formulas of noisy XOR and AND gates.
