We prove that QCSP$(\mathbb{N};x=y\rightarrow y=z)$ is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP$(Γ)$, for $Γ$ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

翻译：我们证明 QCSP$(\mathbb{N};x=y\rightarrow y=z)$ 是 PSpace-完全的，解决了十余年悬而未决的问题。这完成了等式语言上 QCSP 的复杂性分类，即 Logspace、NP-完全与 PSpace-完全三者间的三分法。此外，我们还确定了有界交替 QCSP$(Γ)$ 的分类，其中 $Γ$ 为等式语言。此类问题要么属于 Logspace，要么是 NP-完全或 co-NP-完全，要么在多项式层级中呈现复杂度上升。