Independence testing plays a crucial role in many statistical methodologies, including causal discovery, feature selection, sensitivity analysis, independent component analysis and many others. When the data are independent and identically distributed, kernel methods, particularly those based on the Hilbert-Schmidt independence criterion, stand out for their ability to easily detect nonlinear dependency patterns. They are also highly valued for the strong theoretical guarantees they provide. For time series data, things are more complicated: a new body of theoretical results is derived under strong assumptions about the nature of the underlying random processes, and all test procedures must be revised accordingly to maintain an equivalent level of performance. Shift-HSIC is often considered as a state-of-the-art kernel-based independence test for time series data, but its algorithmic complexity is prohibitive in many situations. In this paper, we propose two low-complexity variants of Shift-HSIC. Both demonstrate strict control over Type-I error and remarkable detection power, while significantly reducing the computational load.