Abstract: The asymmetric quantum Cournot game model under relative profit maximization is analyzed, of which the aim is to study complex behaviors and decision-making processes in economic systems. The concept of quantum entanglement is introduced, and in combination with asymmetric demand functions and cost functions, the dynamic interactions and strategy choices between enterprises are explored. By establishing a dynamic game model, the evolution process and stability conditions of the system are revealed. The effects of quantum entanglement and asymmetric demand and cost functions on the stability of the Nash equilibrium point and complex dynamic behavior are examined. Numerical simulations show that when other parameters are fixed, a smaller adjustment speed can keep the market environment relatively stable. Quantum entanglement can enhance the stability of the system. When the company's output adjustment speed is too large, it will cause the system to exhibit complex chaotic characteristics, and quantum entanglement can effectively prevent the emergence of chaotic states. Finally, the effects of asymmetry on equilibrium solutions and stability are compared.