Superconstant lower bounds for distributed LLL and sinkless orientation in quantum-LOCAL

Establish superconstant lower bounds on the round complexity of sinkless orientation and the distributed Lovász Local Lemma in the quantum-LOCAL model and in stronger locality models.

Background

Prior to this work, no superconstant lower bounds were known for the distributed Lovász Local Lemma or sinkless orientation in the quantum-LOCAL model (or stronger models), and obtaining even large constant lower bounds seemed out of reach of existing techniques. The authors highlight that this gap had been explicitly identified as an open problem.

The paper develops a new lower bound technique and ultimately proves a 2{Ω(log* n)} lower bound for sinkless orientation in the randomized online-LOCAL model, implying the same lower bounds for LLL and across several stronger models, including quantum-LOCAL. This addresses the stated open problem and provides a foundation for broader post-quantum lower bound techniques.

References

In fact, even obtaining large constant lower bounds for sinkless orientation (and therefore also the distributed LLL) in quantum-LOCAL and stronger models seemed out of reach of current techniques and was stated as an open problem in [suomela2024open]. We obtain the following fundamental open problem. Prove superconstant lower bounds for the distributed LLL and sinkless orientation in the quantum-LOCAL model (and stronger models).

— A Post-Quantum Lower Bound for the Distributed Lovász Local Lemma (2510.15698 - Brandt et al., 17 Oct 2025) in Open Problem 2 (op2), Section 1 (Introduction), subsection “Lower bounds for the distributed LLL and sinkless orientation in quantum-LOCAL”