The Importance of Phase in Quantum Mechanics
Although when finally measuring, the phase does not play an impact on the probability of measuring a state, it does play an impact in the quantum circuit before measurement when quantum gates are applied. While an entangled state is going through a quantum circuit, its phase is affected by its interactions with the gates and can lead to different probabilities. For example, and when measured have the same 50-50 probability of measuring or (the phase does not impact measurement), but in and , the phase does impact the final measurement.
The phase stores extra information, which can be extracted through gates like the Hadamard gate.