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In this article, the issue of the finite-time anti-synchronization and fixed-time quasi-anti-synchronization for complex-valued neural networks with time-varying delays and leakage delay is explored. First of all, the model is divided into two equivalent real-valued neural networks employing the decomposing technique. Second, by utilizing the Hölder inequality, constructed function and new designed quantized controller, several novel sufficient criteria are obtained to ensure finite-time anti-synchronization of the studied system. In addition, a feedback control strategy is developed to derive some conclusions of quasi-anti-synchronization by applying the Lyapunov function. Meanwhile, the estimated settling time is acquired which does not rely on the initial value. Two methods our paper used to analyze can provide new ideas for future research. Finally, numerical simulation with simulation results is offered to indicate the efficiency and validity of the proposed theoretical results, and the results are applied to the field of image encryption. [ABSTRACT FROM AUTHOR]