Sample Entropy (SampEn) is widely used to assess the complexity of physiological time-series signals. However, it is a computationally intensive algorithm with O(N2) time complexity. Although algorithmic optimizations, such as Bucket-Assisted SampEn, have been proposed to eliminate unnecessary computations, the time demand restricts their use in real-time applications with long-term inputs. To address the time and space complexity issue in SampEn, we optimize Bucket-Assisted SampEn by dynamic memory allocation to avoid space complexity and accelerate the optimized Bucket-Assisted SampEn on Field Programmable Gate Arrays (FPGA). Our method accelerates Bucket-Assisted SampEn through efficient random storage and data access on FPGA. Furthermore, we introduce a scheduling strategy to handle unbalanced loads for time-intensive inter- and intra-similarity comparisons. We validate our approach on multi-source biomedical signals and demonstrate its effectiveness by achieving more than two orders of magnitude faster than straightforward defined SampEn computation. Our work provides a practical and effective approach for measuring time series complexity using Bucket-Assisted SampEn on FPGA, with the potential for real-time applications with long-term inputs.