Large-scale human mobility data (e.g. GPS data) encodes valuable information interested by various fields. Extracting stay and movement behaviors from noisy positioning record sequences is a critical preliminary step to utilize human mobility data. For the past two decades, this processing has been founded on a simple intuition proposed by Hariharan and Zheng et al.[18, 48], which uses manually selected parameters to make recursive, rule-based classification as to whether data points in a positioning record sequence constitute noise, move, or stay. This de facto processing approach, despite its simplicity, is inherently sensitive to parameter choice and suffers from the low efficiency of sequential processing. These inherent limitations make it practically infeasible, when confronted with the large-scale, fine-grained human mobility datasets in industry. To address this fundamental problem in human mobility data utilization, we innotatively rethink the distinction in representation patterns of noise/stay/move within the positioning record sequence from the lens of topological persistence, culminating in a novel pipeline for robust and efficient human mobility data processing. This is grounded in our empirical observation that topological persistence features of stay/move/noise exhibit robust and generalizable discriminability across variations in parameter choice, individuals, and geographical regions. By introducing the Laplacian to simplify the computation of topological persistence features, our processing pipeline is capable to exploit GPUs' parallel capacity for efficient processing. Experiments on real-world GPS datasets totaling up to thousand billion data points demonstrate that our method produces processing results comparable to those of human annotators, while requiring only 10% of the time consumed by previous approaches. We further show that our method is scalable for cumulative data volume and remains effective in identifying stay/move behaviors that traditional techniques consistently fail to handle, even under conditions of severe positioning errors and diverse behavioral patterns.