  We describe a general technique that yields the first {\em Statistical Query
lower bounds} for a range of fundamental high-dimensional learning problems
involving Gaussian distributions. Our main results are for the problems of (1)
learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of
a single unknown Gaussian distribution. For each of these problems, we show a
{\em super-polynomial gap} between the (information-theoretic) sample
complexity and the computational complexity of {\em any} Statistical Query
algorithm for the problem. Our SQ lower bound for Problem (1) is qualitatively
matched by known learning algorithms for GMMs. Our lower bound for Problem (2)
implies that the accuracy of the robust learning algorithm
in~\cite{DiakonikolasKKLMS16} is essentially best possible among all
polynomial-time SQ algorithms.
  Our SQ lower bounds are attained via a unified moment-matching technique that
is useful in other contexts and may be of broader interest. Our technique
yields nearly-tight lower bounds for a number of related unsupervised
estimation problems. Specifically, for the problems of (3) robust covariance
estimation in spectral norm, and (4) robust sparse mean estimation, we
establish a quadratic {\em statistical--computational tradeoff} for SQ
algorithms, matching known upper bounds. Finally, our technique can be used to
obtain tight sample complexity lower bounds for high-dimensional {\em testing}
problems. Specifically, for the classical problem of robustly {\em testing} an
unknown mean (known covariance) Gaussian, our technique implies an
information-theoretic sample lower bound that scales {\em linearly} in the
dimension. Our sample lower bound matches the sample complexity of the
corresponding robust {\em learning} problem and separates the sample complexity
of robust testing from standard (non-robust) testing.
