  Quantum pre- and post-selection (PPS) paradoxes occur when counterfactual
inferences are made about different measurements that might have been
performed, between two measurements that are actually performed. The 3 box
paradox is the paradigm example of such a paradox, where a ball is placed in
one of three boxes and it is inferred that it would have been found, with
certainty, both in box 1 and in box 2 had either box been opened on their own.
Precisely what is at stake in PPS paradoxes has been unclear, and classical
models have been suggested which are supposed to mimic the essential features
of the problem. We show that the essential difference between the classical and
quantum pre- and post-selection effects lies in the fact that for a quantum PPS
paradox to occur the intervening measurement, had it been performed, would need
to be invasive but non-detectable. This invasiveness is required even for null
result measurements. While some quasi-classical features (such as
non-contextuality and macrorealism) are compatible with PPS paradoxes, it seems
no fully classical model of the 3 box paradox is possible.
