  Estimating correctly the quantum phase of a physical system is a central
problem in quantum parameter estimation theory due to its wide range of
applications from quantum metrology to cryptography. Ideally, the optimal
quantum estimator is given by the so-called quantum Cram\'er-Rao bound, so any
measurement strategy aims to obtain estimations as close as possible to it.
However, more often than not, the current state-of-the-art methods to estimate
quantum phases fail to reach this bound as they rely on maximum likelihood
estimators of non-identifiable likelihood functions. In this work we thoroughly
review various schemes for estimating the phase of a qubit, identifying the
underlying problem which prohibits these methods to reach the quantum
Cram\'er-Rao bound, and propose a new adaptive scheme based on covariant
measurements to circumvent this problem. Our findings are carefully checked by
Monte Carlo simulations, showing that the method we propose is both
mathematically and experimentally more realistic and more efficient than the
methods currently available.
