Break the log-squared space barrier for polynomial-time TreeEval (standard space)

Determine whether the Tree Evaluation Problem admits a polynomial-time algorithm that uses o(log^2 n) space in the standard (free) space model, i.e., whether there exists a polynomial-time procedure for TreeEval that uses less than Ω(log^2 n) space.

Background

A key challenge in evaluating TreeEval efficiently is overcoming the apparent Ω(log2 n) space barrier suggested by pebbling-based strategies in the canonical setting (binary trees of height log n).

Henzinger, Pătraşcu, and Rubinfeld (as cited) explicitly posed whether one can design a polynomial-time algorithm using less than Ω(log2 n) space as an open question, and then studied a different problem focused on catalytic space with limited free space.

The present paper achieves polynomial time with O(log n) free space and O(log{1+ε} n) catalytic space, giving total space below log2 n when catalytic space is allowed. However, the question as posed for standard (free) space remains open.

References

Recently, asked whether a polynomial-time procedure can use less than $\Omega(\log2 n)$ space (stated as an open question), and they obtained results about a different question:

— Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling  (2604.02606 - Asadi et al., 3 Apr 2026) in Section 1: Introduction and statement of results