Exact complexity of SSQR and USSR

Determine the exact computational complexity of the sum-of-square-roots problem (SSQR) and of the unary variant USSR, including whether either problem is solvable in polynomial time or complete for a known class.

Background

The SSQR problem asks whether ∑_{i=1}n a_i{1/2} ≥ k for positive integers a_i and k; the unary variant USSR restricts inputs to unary. Both lie in ER, with current upper bounds placing SSQR in the third level of the counting hierarchy via PosSLP, but neither is known to lie in NP or P.

The authors emphasize that resolving these problems would clarify gaps between word RAM and real RAM models, and that several geometric optimization problems reduce to SSQR in various forms.

References

It is quite possible that USSR\ and SSQR\ are polynomial-time solvable, and many researchers believe so, but their exact complexity remains open.

— The Existential Theory of the Reals as a Complexity Class: A Compendium  (2407.18006 - Schaefer et al., 2024) in Section 'Inside ER'

A possible certificate route to MA\cap\mathsf{coMA} remains unresolved. Algebraic sums-of-squares existence is insufficient: the numerators and the identities to be checked need uniformly short, coherent descriptions.

— $\exists \mathbb{R} \subseteq \textsf{CH}$  (2610.10514 - Meiburg, 7 Oct 2026) in Section 4.2, subsection “Square-root sum”