The quantum smooth label cover problem is undecidable
Abstract
We show that the quantum smooth label cover problem is undecidable and RE-hard. This sharply contrasts the quantum unique label cover problem, which can be decided efficiently by a result of Kempe, Regev, and Toner (FOCS'08). On the other hand, our result aligns with the RE-hardness of the quantum label cover problem, which follows from the celebrated MIP* = RE result of Ji, Natarajan, Vidick, Wright, and Yuen (ACM'21). Additionally, we show that the quantum oracularized smooth label cover problem is RE-hard. Our second result fits with the alternative quantum unique games conjecture recently proposed by Mousavi and Spirig (ITCS'25) on the RE-hardness of the quantum oracularized unique label cover problem. Our proof techniques include a quantum version of Feige's reduction from 3SAT to 3SAT5 (STOC'96) for BCSMIP*-protocols, which may be of independent interest.
Cite
@article{arxiv.2510.03477,
title = {The quantum smooth label cover problem is undecidable},
author = {Eric Culf and Kieran Mastel and Connor Paddock and Taro Spirig},
journal= {arXiv preprint arXiv:2510.03477},
year = {2025}
}
Comments
Revised introduction, incorporated parallel repetition for projection games into Lemma 12