In the Cournot duopoly market, each firm is not completely rational, one firm's possession of information is not symmetrical, and their Nash equilibria are also usually inferior to Pareto optimality. The intersection of quantum game theory, nonlinear dynamics and asymmetric information may be an effective way to solve this problem. In this paper, we set up a multi-period quantum Cournot duopoly game model with bounded rationality based on asymmetric information, investigate the effects of quantum entanglement and asymmetric information on the local stability of quantum Nash equilibrium and complex dynamic behavior of the system. The result shows that an increase in the quantum entanglement decreases the quantum Nash equilibrium local stability and accelerates the occurrence of the system’s chaotic behavior, while the asymmetric information stabilizes the quantum Nash equilibrium and suppresses the chaotic behavior.The contradictive impact on the local stability of quantum Nash equilibrium creates the interesting properties of the game. Numerical simulations demonstrate these complex dynamical features via bifurcation, strange attractors, fractal dimensions and sensitivity to initial conditions.