Persistence of a strict quantum advantage for higher-cardinality updates

Determine whether, for integers t≥2 and dimensions d>2t in the balanced hypercube rendezvous problem, the quantum value ω_Q(d,t) is strictly greater than the classical value ω_C(d,t).

Background

For nearest-neighbor updates (t=1), the paper establishes quantum advantage whenever a real Hadamard matrix of order d exists and d is not a power of two: the quantum value reaches the no-signalling benchmark, while the classical value does not. For t≥2, the paper proves a finite-dimensional obstruction to projective t-sphere resolutions and therefore rules out exact finite-dimensional quantum attainment of the no-signalling value.

It remains unresolved whether quantum strategies nevertheless outperform classical strategies in the higher-cardinality regime. The question is logically distinct from the possible strict gap below the no-signalling benchmark: quantum and classical values could be separated even if neither reaches the benchmark, or they could coincide.