We introduce a methodology and investigate the feasibility of measuring quantum properties of tau lepton pairs in the H→τ+τ− decay at future lepton colliders. In particular, observation of entanglement, steerability and violation of Bell inequalities are examined for the ILC and FCC-ee. We find that detecting quantum correlation crucially relies on precise reconstruction of the tau leptons rest frame and a simple kinematics reconstruction does not suffice due to the finite energy resolution of the colliding beams and detectors. To correct for energy mismeasurements, a log-likelihood method is developed that incorporates the information of impact parameters of tau lepton decays. We demonstrate that an accurate measurement of quantum properties is possible with this method. As a by-product, we show that a novel model-independent test of CP violation can be performed and the CP-phase of Hττ interaction can be constrained with an accuracy comparable to dedicated analyses, i.e., up to 7.9∘ and 5.4∘ at ILC and FCC-ee, respectively.