Contextuality and Bell nonlocality

7 problems

• Is contextuality the resource behind quantum computational speedup?

  Contested

  For quantum computers that draw their power from special 'magic' input states, contextuality was proved necessary for universal computing when the basic units have an odd prime number of levels. Whether this holds for ordinary qubits and other models is disputed.

• Is the $I3322$ quantum maximum reached in finite dimension?

  Partially resolved

  $I3322$ is the simplest Bell test beyond the standard one, with three measurement choices per side. Numerics suggested that reaching its quantum maximum needs systems with infinitely many levels; two September 2026 preprints claim proofs that await independent checking.

• Is every entangled state nonlocal given many copies?

  Open

  Some entangled states never violate a Bell inequality when measured one pair at a time. The question is whether measuring many copies together, with local filtering, always reveals nonlocality for every entangled state.

• Smallest Kochen-Specker vector set in three dimensions

  Partially resolved

  The Kochen-Specker theorem uses a finite set of directions in 3D space to prove that quantum outcomes cannot all be fixed in advance. The smallest such set has between 24 and 31 directions, and the exact number is unknown.

• Explicit Bell game separating commuting-operator and tensor-product quantum models

  Partially resolved

  Quantum correlations can be modeled with a separate space for each lab or with commuting operators on one shared space. A 2020 result proved these give different correlations, but no explicit example is known.

• Exact nonlocality threshold of two-qubit Werner states

  Partially resolved

  A Werner state mixes a perfectly entangled pair with random noise. Below some noise level all projective-measurement results can be reproduced by classical hidden variables, and that exact level is unknown.

• Do general measurements beat projective ones for Werner-state nonlocality?

  Open

  Measurements can be generalized beyond simple projections by using helper systems. For Werner states it is unknown whether such generalized measurements reveal nonlocality at higher noise than projective ones.