Discovering the underlying Directed Acyclic Graph (DAG) from time series observational data is highly challenging due to the dynamic nature and complex nonlinear interactions between variables. Existing methods typically search for the optimal DAG by optimizing an objective function but face scalability challenges, as their computational demands grow exponentially with the dimensional expansion of variables. To this end, we propose LOCAL, a highly efficient, easy-to-implement, and constraint-free method for recovering dynamic causal structures. LOCAL is the first attempt to formulate a quasi-maximum likelihood-based score function for learning the dynamic DAG equivalent to the ground truth. Building on this, we introduce two adaptive modules that enhance the algebraic characterization of acyclicity: Asymptotic Causal Mask Learning (ACML) and Dynamic Graph Parameter Learning (DGPL). ACML constructs causal masks using learnable priority vectors and the Gumbel-Sigmoid function, ensuring DAG formation while optimizing computational efficiency. DGPL transforms causal learning into decomposed matrix products, capturing dynamic causal structure in high-dimensional data and improving interpretability. Extensive experiments on synthetic and real-world datasets demonstrate that LOCAL significantly outperforms existing methods and highlight LOCAL's potential as a robust and efficient method for dynamic causal discovery.