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Temporal logics for the specification of information-flow properties are ableto express relations between multiple executions of a system. The two mostimportant such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL*by trace quantification. It is known that this expressiveness comes at a price,i.e. satisfiability is undecidable for both logics.  In this paper we settle the exact complexity of these problems, showing thatboth are in fact highly undecidable: we prove that HyperLTL satisfiability is$\Sigma_1^1$-complete and HyperCTL* satisfiability is $\Sigma_1^2$-complete.These are significant increases over the previously known lower bounds and thefirst upper bounds. To prove $\Sigma_1^2$-membership for HyperCTL*, we provethat every satisfiable HyperCTL* sentence has a model that is equinumerous tothe continuum, the first upper bound of this kind. We prove this bound to betight. Finally, we show that the membership problem for every level of theHyperLTL quantifier alternation hierarchy is $\Pi_1^1$-complete.

HyperLTL Satisfiability is $\Sigma_1^1$-complete, HyperCTL* Satisfiability is $\Sigma_1^2$-complete