The multiple-quantum NMR spectroscopy has an extensive application in determination of the bio-macro-molecular structures and in the investigation of the properties of a variety of physical materials. In quantum computation the multiple-quantum transition processes have been used to construct the quantum circuits, quantum algorithms, and quantum simulations. The multiple-quantum operator algebra spaces are closely related to the symmetries of a multiple-spin quantum system. They may have an important effect on the multiple-quantum transition processes and the multiple-quantum NMR spectroscopy of the spin system. Here gives a brief introduction to the multiple-quantum operator algebra spaces.