Determine whether the GHZ Hypergraph Hamiltonian exhibits a quantum overlap gap property

Determine whether the GHZ Hypergraph Hamiltonian has a Quantum Overlap Gap Property or whether its near-optimal solution space lacks the clustering structure required for such a property.

Background

The appendix studies an EPR/GHZ-based generalization of the Quantum Max-Cut Hamiltonian and derives upper and lower bounds on its maximal energy. For large hyperedge size k, these bounds nearly coincide and are attained by low-entropy configurations, which appears to obstruct the clustering phenomenon underlying the Quantum Overlap Gap Property.

The authors explicitly state that they do not formally establish non-existence of the QOGP for this Hamiltonian. Consequently, whether a QOGP exists remains unresolved; the possibility of average-case efficient quantum algorithms for the GHZ Hypergraph Hamiltonian is also left open for large k.