since the dimensions of formula_118 and formula_119 are, in principle, unbounded, determining whether a given box formula_44 admits a quantum realization is a complicated problem. in fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. moreover, the problem of deciding whether formula_44 can be approximated by a quantum system with precision formula_122 is np-hard. characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints.