  A formula for the commutator of tensor product matrices is used to shows
that, for qubits, compatibility of quantum multiparty observables almost never
implies local compatibility at each site and to predict when this happens/does
not happen in a concise manner. In particular, it is shown that two ``fully
nontrivial'' $n$-qubit observables are compatible locally and globally if and
only if they are equal up to sign. In addition, the formula gives insight into
the construction of new paradoxes of the type of the Kochen-Specker Theorem,
which can then be easily rephrased into proposals for new no hidden variable
experiments of the type of the ``Bell Theorem without inequalities''.
