  Complementarity is one of the main features of quantum physics that radically
departs from classical notions. Here we consider the limitations that this
principle imposes due to the unpredictability of measurement outcomes of
incompatible observables. For two-level systems, it is shown that any
preparation violating complementarity enables the preparation of a
non-signalling box violating Tsirelson's bound. Moreover, these
"beyond-quantum" objects could be used to distinguish a plethora of
non-orthogonal quantum states and hence enable improved cloning protocols. For
higher-dimensional systems the main ideas are sketched.
