Temporal correlations are a family of things, since there are many degrees of freedom to choose what to compare with what.
A very simple test is the "Granger Causality", which tests whether one timeline significantly predicts another one. You can formulate more complex models such as time series a predicting b with a certain lag.
Ultimately, the idea is most often to remove unrelated factors (such as control variables, seasonality, self-influence i.e. autoregression) and then measure how well one series at t(0) predicts another one at t(1), while optionally doing some sort of hyperparameter optimization for the lag (i.e. determine which lag works best).
A very simple test is the "Granger Causality", which tests whether one timeline significantly predicts another one. You can formulate more complex models such as time series a predicting b with a certain lag.
Ultimately, the idea is most often to remove unrelated factors (such as control variables, seasonality, self-influence i.e. autoregression) and then measure how well one series at t(0) predicts another one at t(1), while optionally doing some sort of hyperparameter optimization for the lag (i.e. determine which lag works best).