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Oblivious polynomial evaluation (OPE) is a secure two-party cryptographic protocol that enables a receiver to obliviously retrieve polynomial value on its private input for a private polynomial of a sender. This paper deals with the problem of OPE in the quantum domain. Herein, we propose the construction of the first quantum secure OPE, namely QuOPE. Our design is a constant round protocol that only requires quantum communication of one quantum superposition state that achieves all the stipulated security requirements of an OPE. In particular, privacy and security of the sender and receiver's private data are ensured. In addition, we extend the idea of QuOPE to quantum rational polynomial evaluation (QuORPE). Furthermore, we explore the possibility of utilizing QuOPE as a building block in the automated medical diagnosis platform. [ABSTRACT FROM AUTHOR]