  Boolean networks are a general model of interacting entities, with
applications to biological phenomena such as gene regulation. Attractors play a
central role, and the schedule of entities update is a priori unknown. This
article presents results on the computational complexity of problems related to
the existence of update schedules such that some limit-cycle lengths are
possible or not. We first prove that given a Boolean network updated in
parallel, knowing whether it has at least one limit-cycle of length $k$ is
$\text{NP}$-complete. Adding an existential quantification on the
block-sequential update schedule does not change the complexity class of the
problem, but the following alternation brings us one level above in the
polynomial hierarchy: given a Boolean network, knowing whether there exists a
block-sequential update schedule such that it has no limit-cycle of length $k$
is $\Sigma_2^\text{P}$-complete.
