We survey complexity classifications (classical and recent) for different computational
problems for Boolean constraints. Among the problems addressed are the satisfiability
problem for Boolean constraints, the evaluation problem for quantified constraints, both
with and without bound on the number of quantifier alternations, the problem to count the
number of solutions for quantified constraints, again with and without bounding quantifier
alternations. Particular attention is paid to the question of how much the theory of Boolean
clones and Post's lattice can be used to obtain the desired classifications. While for all
mentioned problems the classification is obtained making heavy use of the structure of the
lattice (``the Galois connection holds a priori''), other problems are pointed out for which we
only notice after obtaining the classification by different means that it actually obeys the border
of Boolean relational clones (``the Galois connection holds a posteriori'') or for which the
classifaction cuts across the border of relational clones (``the Galois connection does not
hold''). Finally, a further question is addressed in which the lattic of relational clones is only of
limited use; this concerns a fine classification of the complexity of the satisfiability problem for
Boolean CSPs using very strict reductions such as FO.