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A remarkable feature of quantum theory is non-locality (i.e. the presence ofcorrelations which violate Bell inequalities). However, quantum correlationsare not maximally non-local, and it is natural to ask whether there arecompelling reasons for rejecting theories in which stronger violations arepossible. To shed light on this question, we consider post-quantum theories inwhich maximally non-local states (non-local boxes) occur. It has previouslybeen conjectured that the set of dynamical transformations possible in suchtheories is severely limited. We settle the question affirmatively in the caseof reversible dynamics, by completely characterizing all such transformationsallowed in this setting. We find that the dynamical group is trivial, in thesense that it is generated solely by local operations and permutations ofsystems. In particular, no correlations can ever be created; non-local boxescannot be prepared from product states (in other words, no analogues ofentangling unitary operations exist), and classical computers can efficientlysimulate all such processes.

All reversible dynamics in maximally non-local theories are trivial