The precise measurement of the leptonic CP-violating phase δCP remains one of the major open challenges in neutrino physics, as current experiments achieve only very limited accuracy. We address this issue through the lens of quantum estimation theory. A distinctive feature of neutrino oscillation experiments is that they cannot freely optimize the probe or measurement, since both are constrained by the production and detection of flavor eigenstates. We therefore examine whether the large uncertainty in δCP originates from intrinsic reasons, either of the neutrino quantum state or of flavor measurements, or if it instead stems from experimental limitations. By comparing quantum and classical Fisher information, we demonstrate that the limited sensitivity to δCP originates primarily from the information content of flavor measurements. Furthermore, we show that targeting the second oscillation maximum, as in the ESSνSB proposal, substantially enhances δCP information compared to experiments centered on the first maximum.