Separations in the polynomial hierarchy

Ascertain whether the polynomial hierarchy separates at each level; specifically, prove or refute for k ∈ ℕ that Σ_k^p ≠ Σ_{k+1}^p (and analogously Π_k^p ≠ Π_{k+1}^p), or determine whether the hierarchy collapses at some level.

Background

In discussing computational complexity background relevant to bilevel optimization hardness, the paper recalls a central open question in complexity theory: whether the polynomial hierarchy is strict at each level or collapses.

This question underpins the interpretation of Σ_2p-hardness results, since separations between levels imply that such problems are strictly harder than NP-complete problems.

References

For any $ k\in \NN$, it remains unknown whether $\Sigma_kp \neq \Sigma_{k+1}p$ or $\Sigma_kp = \Sigma_{k+1}p$, and similarly for $\Pi_{k}p$ and $\Pi_{k+1}p$.

— Geometric and computational hardness of bilevel programming  (2407.12372 - Bolte et al., 2024) in Section 4.1 (Polynomial hierarchy)

The complexity class $\Sigma_2p$ contains the well-known complexity class \NP \ (i.e., $\NP \subseteq \Sigma_2p$), but it is an open problem whether proper inclusion holds or not.

— Vertex Cover Interdiction in Bipartite Graphs  (2609.24624 - Ito et al., 21 Sep 2026) in Section Introduction