  Quantum three box paradox is a prototypical example of some bizarre
predictions for intermediate measurements made on pre- and post-selected
systems. Although in principle those effects can be explained by measurement
disturbance, it is not clear what mechanisms are required to fully account for
the observed correlations. In this paper, this paradox is scrutinised from the
causal point of view. We consider an array of potential causal structures
behind the experiment, eliminating those without enough explanatory power. This
gives a means of differentiating between the various mechanisms in which
measurement disturbance can propagate in the system. Specifically, we
distinguish whether it is just the measurement outcome or the full measurement
context that is required for the causal explanation of the observed statistics.
We show that the latter is indispensable, but only when the full statistics is
taken into account (which includes checking the third box too). Furthermore, we
discuss the realism assumption which posits the existence of preexisting values
revealed by measurements. It is shown that in this case measurement disturbance
is necessary. Interestingly, without the realism assumption, the original
version of the paradox (with just two boxes considered for inspection) can be
explained without resorting to any measurement disturbance. These various
results illustrate the richness of the paradox which is better appreciated from
the causal perspective.
