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Nisan–Wigderson weaker hypothesis on time versus Σ2 time

Determine whether there exists an ε > 0 such that TIME[t(n)] is not contained in Σ₂TIME[t(n)^{1−ε}], i.e., prove the weaker Nisan–Wigderson hypothesis that for some ε > 0, TIME[t(n)] ⊄ Σ₂TIME[t(n)^{1−ε}].

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Background

The Nisan–Wigderson derandomization can be based on a hypothesis comparing deterministic time classes to second-level time with alternations, asserting a non-containment with a sublinear exponent.

The paper notes this hypothesis remains open and suggests current techniques in the work likely cannot refute it, underscoring its status as a robust open problem in complexity theory.

References

Note that the derandomization of Nisan and Wigderson can also be based on assuming the weaker hypothesis $\TIME[t] \not\subset \Sigma_2 \TIME[t{1-\eps}]$, which remains open and almost certainly cannot be refuted using the ideas of this paper.

— Simulating Time With Square-Root Space (2502.17779 - Williams, 25 Feb 2025) in Footnote in Introduction (following discussion of Nisan–Wigderson derandomization)