A Counterexample to the Generalized Linial-Nisan Conjecture

TL;DR

This paper disproves the Generalized Linial-Nisan Conjecture by providing a counterexample for circuits of depth 3 or higher, impacting complexity theory and the understanding of circuit indistinguishability.

Contribution

It presents the first known counterexample to the GLN Conjecture, showing that almost k-wise independent distributions can be distinguished by certain circuits, challenging previous assumptions.

Findings

Counterexample invalidates the GLN Conjecture for depth 3+ circuits.

Implication that Pi2P is not contained in P^NP relative to a random oracle.

Shows limitations of previous results on AC0 functions structure.

Abstract

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth circuits. The original Linial-Nisan Conjecture was recently proved by Braverman; we offered a $200 prize for the generalized version. In this paper, we save ourselves $200 by showing that the GLN Conjecture is false, at least for circuits of depth 3 and higher. As a byproduct, our counterexample also implies that Pi2P is not contained in P^NP relative to a random oracle with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the highest levels of PH previously proved separate were NP and coNP. Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC0 functions, cannot be improved in several interesting respects.