  We study $q$-SAT in the multistage model, focusing on the linear-time
solvable 2-SAT. Herein, given a sequence of $q$-CNF fomulas and a non-negative
integer $d$, the question is whether there is a sequence of satisfying truth
assignments such that for every two consecutive truth assignments, the number
of variables whose values changed is at most $d$. We prove that Multistage
2-SAT is NP-hard even in quite restricted cases. Moreover, we present
parameterized algorithms (including kernelization) for Multistage 2-SAT and
prove them to be asymptotically optimal.
