Position of ER relative to the polynomial-time hierarchy

Determine the position of the complexity class Exists-R (ER) relative to the polynomial-time hierarchy (PH), and ascertain whether NP equals ER or whether ER is contained in or separates from PH.

Background

The survey recalls the few known relationships between ER and classical classes: NP ⊆ ER via direct ETR encodings, and ER ⊆ PSPACE via Canny's roadmap method. Beyond these inclusions, the authors note a lack of knowledge about ER’s placement among traditional classes and highlight PH in particular as unresolved.

They also remark that while NP = ER cannot be excluded, most researchers consider it unlikely; nevertheless, no structural separations or complete characterizations are currently available.

References

And this is all we know about with respect to traditional complexity classes. The status of with respect to for instance, is wide open. The possibility that NP = cannot yet be ruled out, but is usually considered unlikely.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Section 'Where is the Existential Theory of the Reals?'