  An important problem in quantum information theory is that of bounding sets
of correlations that arise from making local measurements on entangled states
of arbitrary dimension. Currently, the best-known method to tackle this problem
is the NPA hierarchy; an infinite sequence of semidefinite programs that
provides increasingly tighter outer approximations to the desired set of
correlations. In this work we consider a more general scenario in which one
performs sequences of local measurements on an entangled state of arbitrary
dimension. We show that a simple adaptation of the original NPA hierarchy
provides an analogous hierarchy for this scenario, with comparable resource
requirements and convergence properties. We then use the method to tackle some
problems in device-independent quantum information. First, we show how one can
robustly certify over 2.3 bits of device-independent local randomness from a
two-quibt state using a sequence of measurements, going beyond the theoretical
maximum of two bits that can be achieved with non-sequential measurements.
Finally, we show tight upper bounds to two previously defined tasks in
sequential Bell test scenarios.
