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This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. 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