  Many problems in quantum information theory can be formulated as
optimizations over the sequential outcomes of dynamical systems subject to
unpredictable external influences. Such problems include many-body entanglement
detection through adaptive measurements, computing the maximum average score of
a preparation game over a continuous set of target states and limiting the
behavior of a (quantum) finite-state automaton. In this work, we introduce
tractable relaxations of this class of optimization problems. To illustrate
their performance, we use them to: (a) compute the probability that a
finite-state automaton outputs a given sequence of bits; (b) develop a new
many-body entanglement detection protocol; (c) let the computer invent an
adaptive protocol for magic state detection. As we further show, the maximum
score of a sequential problem in the limit of infinitely many time steps is in
general incomputable. Nonetheless, we provide general heuristics to bound this
quantity and show that they provide useful estimates in relevant scenarios.
