  When conducting a Bell test, it is normal to assume that the preparation of
the quantum state is independent of the measurements performed on it.
Remarkably, the violation of local realism by entangled quantum systems can be
certified even if this assumption is partially relaxed. Here, we allow such
measurement dependence to correlate multiple runs of the experiment, going
beyond previous studies that considered independent and identically distributed
(i.i.d.) runs. To do so, we study the polytope that defines block-i.i.d.
measurement-dependent local models. We prove that non-i.i.d. models are
strictly more powerful than i.i.d. ones, and comment on the relevance of this
work for the study of randomness amplification in simple Bell scenarios with
suitably optimised inequalities.
