On beating the hybrid argument

The hybrid argument allows one to relate the distinguishability of a distribution (from uniform) to the predictability of individual bits given a prefix. The argument incurs a loss of a factor k equal to the bit-length of the distributions: ε-distinguishability implies ε/k-predictability. This paper studies the consequences of avoiding this loss - what we call "beating the hybrid argument" - and develops new proof techniques that circumvent the loss in certain natural settings. Specifically, we obtain the following results: 1. We give an instantiation of the Nisan-Wigderson generator (JCSS '94) that can be broken by quantum computers, and that is o(1)-unpredictable against AC ⁰. We conjecture that this generator indeed fools AC ⁰. Our conjecture implies the existence of an oracle relative to which BQP is not in the PH, a longstanding open problem. 2. We show that the "INW" generator by Impagliazzo, Nisan, and Wigderson (STOC '94) with seed length O(log n log log n) produces a distribution that is 1/log n-unpredictable against poly-logarithmic width (general) read-once oblivious branching programs. Obtaining such generators where the output is indistinguishable from uniform is a longstanding open problem. 3. We identify a property of functions f, "resamplability," that allows us to beat the hybrid argument when arguing indistinguishability of (Equation Presented) from uniform. This gives new pseudorandom generators for classes such as AC ⁰[p] with a stretch that, despite being sub-linear, is the largest known. We view this as a first step towards beating the hybrid argument in the analysis of the Nisan-Wigderson generator (which applies (Equation Presented) on correlated x ₁,...,x _k) and proving the conjecture in the first item.