Retracted: Two-Player Entangled Games are NP-Hard

Authors Anand Natarajan , Thomas Vidick

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Anand Natarajan
  • Center for Theoretical Physics, MIT, Cambridge, USA
Thomas Vidick
  • Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, USA

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Anand Natarajan and Thomas Vidick. Retracted: Two-Player Entangled Games are NP-Hard. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The article, published on June 4th, 2018 in the CCC 2018 proceedings, has been retracted by agreement between the authors, the editor(s), and the publisher Schloss Dagstuhl / LIPIcs. The retraction has been agreed due to an error in the proof of the main result. This error is carried over from an error in the referenced paper “Three-player entangled XOR games are NP-hard to approximate” by Thomas Vidick (SICOMP ’16). That paper was used in an essential way to obtain the present result, and the error cannot be addressed through an erratum. See Retraction Notice on the last page of the PDF. We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP subseteq MIP^*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Interactive proof systems
  • Theory of computation → Complexity classes
  • low-degree testing
  • entangled nonlocal games
  • multi-prover interactive proof systems


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