It has been challenging to model the complex temporal-spatial dependencies between agents for trajectory prediction. As each state of an agent is closely related to the states of adjacent time steps, capturing the local temporal dependency is beneficial for prediction, while most studies often overlook it. Besides, learning the high-order motion state attributes is expected to enhance spatial interaction modeling, but it is rarely seen in previous works. To address this, we propose a lightweight framework, LTMSformer, to extract temporal-spatial interaction features for multi-modal trajectory prediction. Specifically, we introduce a Local Trend-Aware Attention mechanism to capture the local temporal dependency by leveraging a convolutional attention mechanism with hierarchical local time boxes. Next, to model the spatial interaction dependency, we build a Motion State Encoder to incorporate high-order motion state attributes, such as acceleration, jerk, heading, etc. To further refine the trajectory prediction, we propose a Lightweight Proposal Refinement Module that leverages Multi-Layer Perceptrons for trajectory embedding and generates the refined trajectories with fewer model parameters. Experiment results on the Argoverse 1 dataset demonstrate that our method outperforms the baseline HiVT-64, reducing the minADE by approximately 4.35%, the minFDE by 8.74%, and the MR by 20%. We also achieve higher accuracy than HiVT-128 with a 68% reduction in model size.