The quantum smooth label cover problem is undecidable
Eric Culf, Kieran Mastel, Connor Paddock, Taro Spirig
TL;DR
This work shows that the quantum smooth label cover problem is undecidable and $RE$-hard, contrasting with the known polynomial-time decidability of quantum unique label cover and aligning with the $RE$-hardness phenomena from $MIP^*=RE$. The authors develop a quantum-proof Feige-style reduction from $3 ext{SAT}$ variants to $3 ext{SAT}5^*$ and then to smooth label cover, supported by a weighted-algebras formalism that preserves completeness/soundness through expander-based degree reduction. They also extend the hardness to the quantum oracularized smooth label cover, using a $3 ext{SAT}-10$ construction to enforce 2-oracularizable strategies. Finally, they prove $RE$-hardness for $3 ext{SAT}^*$ with fixed degree via uniform-marginals reductions, expander-graph substitutions, and subdivision arguments, clarifying the landscape of quantum PCP-type hardness and its relation to nonlocal games.
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.