Complexity of SSQR and USSR

Ascertain the exact computational complexity of SSQR (Sum-of-Square-Roots) and USSR (Unary Sum-of-Square-Roots), including whether these problems are solvable in polynomial time.

Background

SSQR asks whether a sum of square roots of integers exceeds a threshold, and USSR is its unary-encoded variant.

Both problems are central to precise numerical comparisons (e.g., Euclidean TSP) and, despite strong upper bounds, their exact placement in standard complexity classes remains unresolved.