  We introduce an extension of Strategy Logic for the imperfect-information
setting, called SLii, and study its model-checking problem. As this logic
naturally captures multi-player games with imperfect information, the problem
turns out to be undecidable. We introduce a syntactical class of "hierarchical
instances" for which, intuitively, as one goes down the syntactic tree of the
formula, strategy quantifications are concerned with finer observations of the
model. We prove that model-checking SLii restricted to hierarchical instances
is decidable. This result, because it allows for complex patterns of
existential and universal quantification on strategies, greatly generalises
previous ones, such as decidability of multi-player games with imperfect
information and hierarchical observations, and decidability of distributed
synthesis for hierarchical systems. To establish the decidability result, we
introduce and study QCTL*ii, an extension of QCTL* (itself an extension of CTL*
with second-order quantification over atomic propositions) by parameterising
its quantifiers with observations. The simple syntax of QCTL* ii allows us to
provide a conceptually neat reduction of SLii to QCTL*ii that separates
concerns, allowing one to forget about strategies and players and focus solely
on second-order quantification. While the model-checking problem of QCTL*ii is,
in general, undecidable, we identify a syntactic fragment of hierarchical
formulas and prove, using an automata-theoretic approach, that it is decidable.
The decidability result for SLii follows since the reduction maps hierarchical
instances of SLii to hierarchical formulas of QCTL*ii .
