New nonparametric methods using local polynomial quantile regression have been developed to test for conditional independence in dependent data. These methods compare estimators that converge under correct conditions and diverge otherwise. The tests can detect local deviations from conditional independence at the same rate as traditional methods. The approach is the first to offer nonparametric tests for time-series conditional independence that can detect local deviations at the standard rate. Simulations show that the tests perform well in practical applications, such as studying the returns to schooling.