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.
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Regaud et al. (2026) studied this question.
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