Based on the Berry-Tabor trace formula, a semiclassical quantization condition for the periodic orbits in a two dimensional uncoupled oscillator system and the correspondence relation between the quantum levels and classical trajectories of the system have been studied in detail. Making use of the quantum-classical correspondence relation, it has been found that if a set of quantum levels corresponds to the periodic orbits with the same topology structure M ( M1, M2 ), there will be long-range correlations among these quantum levels.