The author studies the Hilbert series of the quadratic algebra associated with a unitary \(R\)-matrix. The main result is that the Hilbert series is rational with real positive poles and real negative zeros. This result has also been obtained for a more general \(R\)-matrix-Hecke operator by the reviewer [Acta Math. Vietnam. 24, No. 2, 235-246 (1999; Zbl 0949.16025)] and for a closed Hecke operator, it is shown [\textit{Nguyên Phuong Dung} and the reviewer, Int. Math. Res. Not. 2003, No. 40, 2193-2203 (2003; Zbl 1041.16035)] that the numerator and the denominator of the Poincaré series are reciprocal polynomials. Using the main result, the author gives an example of a Koszul algebra that is not associated to any unitary \(R\)-matrix. He also proposes an application of the main theorem in classifying cotriangular Hopf algebras using a superversion of a theorem of Deligne. Note that this superversion of Deligne's theorem has recently been proved by \textit{P. Deligne} himself [Mosc. Math. J. 2, No. 2, 227-248 (2002; Zbl 1005.18009)]. Note also that the author's idea has also been carried out in more detail by \textit{P. Etingof} and \textit{S. Gelaki} [Am. J. Math. 123, No. 4, 699-713 (2001; Zbl 0990.16030)] using the above mentioned result of Deligne.