  We present an information-theoretic lower bound for the problem of parameter
estimation with time-uniform coverage guarantees. Via a new a reduction to
sequential testing, we obtain stronger lower bounds that capture the hardness
of the time-uniform setting. In the case of location model estimation, logistic
regression, and exponential family models, our $\Omega(\sqrt{n^{-1}\log \log
n})$ lower bound is sharp to within constant factors in typical settings.
