Quantum Random Access Optimizer (QRAO) is a quantum-relaxation based optimization algorithm proposed by Fuller et al. that utilizes Quantum Random Access Code (QRAC) to encode multiple variables of binary optimization in a single qubit. Differing from standard quantum optimizers such as QAOA, it utilizes the eigenstates of local quantum Hamiltonians that are not diagonal in the computational basis. There are indications that quantum entanglement may not be needed to solve binary optimization problems with standard quantum optimizers because their maximal eigenstates of diagonal Hamiltonians include classical states. In this study, while quantumness does not always improve the performance of quantum relaxations, we observed that there exist some instances in which quantum relaxation succeeds to find optimal solutions with the help of quantumness. Our results suggest that QRAO not only can scale the instances of binary optimization problems solvable with limited quantum computers but also can benefit from quantum entanglement.