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Many existed quantum private comparison (QPC) protocols can determine whether two secrets are equal or not, while the quantum private magnitude comparison (QPMC) protocol by Lang can output three results: greater than, equal and less than for two private data. In order to implement the magnitude comparison, it defined the minimum operation. However, if we only rely on this operation to implement QPMC, it may not be efficient at cooperating with some quantum resources. For this reason, it is necessary for us to introduce another operation—maximum one. With regard to some quantum resources, only by using the maximum operation are we able to realize a simple and efficient QPMC. In this paper, it is the maximum operation that helps us utilize a single Bell state to propose a QPMC protocol in an easy and efficient way. The protocol is fully analysed for its correctness and security. The analyses prove that the presented protocol is not only simple yet efficient but also of low costs. It would be a better alternative for QPMC. [ABSTRACT FROM AUTHOR]