The Exponential-Time Hypothesis ( ET H ) is a strengthening of the = conjecture, stating that 3 - SA T on n variables cannot be solved in time 2 n , for some 0"> 0 . In recent years, analogous hypotheses that are ``exponentially-strong'' forms of other classical complexity conjectures (such as or co - ) have also been considered. These Exponential-Time Hypotheses have been widely influential across different areas of complexity theory. However, their connections to *derandomization and circuit lower bounds* have yet to be systematically studied. Such study is indeed the focus of the current work, and we prove a sequence of results demonstrating that *the connections between exponential-time hypotheses, derandomization, and circuit lower bounds are remarkably strong*.

First, we show that if 3 - SA T (or even TQB F ) cannot be solved by probabilistic algorithms that run in time 2 n polylo g ( n ) , then can be deterministically simulated ``on average case'' in (nearly-)polynomial-time (i.e., in time n polyloglo g ( n ) ). This result addresses a long-standing lacuna in uniform ``hardness-to-randomness'' results, which did not previously extend to such parameter settings. Moreover, we extend this result to support an ``almost-always'' derandomization conclusion from an ``almost-always'' lower bound hypothesis.

Secondly, we show that *disproving* certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CircuitSA T for circuits over n bits of size pol y ( n ) can be solved by *probabilistic algorithms* in time 2 n polylo g ( n ) , then does not have circuits of quasilinear size. The main novel feature of this result is that we only assume the existence of a *randomized* circuit-analysis algorithm, whereas previous similar results crucially relied on the hypothesis that the circuit-analysis algorithm does not use randomness.

Thirdly, we show that a very weak exponential-time hypothesis is closely-related to the classical question of whether derandomization and circuit lower bounds are *equivalent*. Specifically, we show two-way implications between the hypothesis that the foregoing equivalence holds and the hypothesis that cannot be decided by ``small'' circuits that are *uniformly generated* by relatively-efficient non-deterministic machines. This highlights a sufficient-and-necessary path for progress towards proving that derandomization and circuit lower bounds are indeed equivalent.