  The problem of measuring a time-varying phase, even when the statistics of
the variation is known, is considerably harder than that of measuring a
constant phase. In particular, the usual bounds on accuracy - such as the
$1/(4\bar{n})$ standard quantum limit with coherent states - do not apply.
Here, restricting to coherent states, we are able to analytically obtain the
achievable accuracy - the equivalent of the standard quantum limit - for a wide
class of phase variation. In particular, we consider the case where the phase
has Gaussian statistics and a power-law spectrum equal to
$\kappa^{p-1}/|\omega|^p$ for large $\omega$, for some $p>1$. For coherent
states with mean photon flux ${\cal N}$, we give the Quantum Cram\'er-Rao Bound
on the mean-square phase error as $[p \sin (\pi/p)]^{-1}(4{\cal
N}/\kappa)^{-(p-1)/p}$. Next, we consider whether the bound can be achieved by
an adaptive homodyne measurement, in the limit ${\cal N}/\kappa \gg 1$ which
allows the photocurrent to be linearized. Applying the optimal filtering for
the resultant linear Gaussian system, we find the same scaling with ${\cal N}$,
but with a prefactor larger by a factor of $p$. By contrast, if we employ
optimal smoothing we can exactly obtain the Quantum Cram{\'e}r-Rao Bound. That
is, contrary to previously considered ($p=2$) cases of phase estimation, here
the improvement offered by smoothing over filtering is not limited to a factor
of 2 but rather can be unbounded by a factor of $p$. We also study numerically
the performance of these estimators for an adaptive measurement in the limit
where ${\cal N}/\kappa$ is not large, and find a more complicated picture.
