A Post-Quantum Lower Bound for the Distributed Lovász Local Lemma

Abstract: In this work, we study the Lovász local lemma (LLL) problem in the area of distributed quantum computing, which has been the focus of attention of recent advances in quantum computing [STOC'24, STOC'25, STOC'25]. We prove a lower bound of $2^{Ω(\log^* n)}$ for the complexity of the distributed LLL in the quantum-LOCAL model. More specifically, we obtain our lower bound already for a very well-studied special case of the LLL, called sinkless orientation, in a stronger model than quantum-LOCAL, called the randomized online-LOCAL model. As a consequence, we obtain the same lower bounds for sinkless orientation and the distributed LLL also in a variety of other models studied across different research communities. Our work provides the first superconstant lower bound for sinkless orientation and the distributed LLL in all of these models, addressing recently stated open questions. Moreover, to obtain our results, we develop an entirely new lower bound technique that we believe has the potential to become the first generic technique for proving post-quantum lower bounds for many of the most important problems studied in the context of locality.