HyperQPTL and HyperQPTL + are expressive specification languages for hyperproperties, properties that relate multiple executions of a system. Tight complexity bounds are known for HyperQPTL finite-state satisfiability and model-checking. Here, we settle the complexity of satisfiability for HyperQPTL as well as satisfiability, finite-state satisfiability, and model-checking for HyperQPTL + : the former is Σ 1 2 -complete, the latter are all equivalent to truth in third-order arithmetic, i.e., all four are very undecidable.

We determine the complexity of second-order HyperLTL satisfiability, finite-state satisfiability, and model-checking: All three are equivalent to truth in third-order arithmetic. We also consider two fragments of second-order HyperLTL that have been introduced with the aim to facilitate effective model-checking by restricting the sets one can quantify over. The first one restricts second-order quantification to smallest/largest sets that satisfy a guard while the second one restricts second-order quantification further to least fixed points of (first-order) HyperLTL definable functions. All three problems for the first fragment are still equivalent to truth in third-order arithmetic while satisfiability for the second fragment is $Σ_1^2$-complete, and finite-state satisfiability and model-checking are equivalent to truth in second-order arithmetic. Finally, we also introduce closed-world semantics for second-order HyperLTL, where set quantification ranges only over subsets of the model, while set quantification in standard semantics ranges over arbitrary sets of traces. Here, satisfiability for the least fixed point fragment becomes $Σ_1^1$-complete, but all other results are unaffected.