We aim to assess the performance of LiDAR-to-map registration on compressive maps. Modern autonomous vehicles utilize pre-built HD (High-Definition) maps to perform sensor-to-map registration, which recovers pose estimation failures and reduces drift in a large-scale environment. However, sensor-to-map registration is usually realized by registering the sensor to a dense 3D model, which occupies massive storage space in the HD map and requires much data processing overhead. Although smaller 3D models are preferable, the optimal compressive map format for preservation of the best registration performance remains unclear. In this paper, we propose a novel and challenging benchmark to evaluate existing LiDAR-to-map registration methods from three perspectives: map compressibility, robustness, and precision. We compared various map formats, including raw points, hierarchical GMMs, and feature points, and show their performance trade-offs between compressibility and robustness on real-world LiDAR datasets: KITTI Odometry Dataset and Argoverse Tracking Dataset. Our benchmark reveals that state-of-the-art deep feature point based methods outperform traditional methods significantly when the map size budget is high. However, when map size budget is low, deep methods are outperformed by the methods using simpler models in Argoverse Tracking Dataset due to poor spatial coverage. In addition, we observe that the recently published TEASER++ significantly outperforms RANSAC for the feature point methods. Our analysis provides a valuable reference for the community to design budgeted real-world systems and find potential research opportunities. We will release the benchmark for public use.