  We consider the observations of an unknown $s$-sparse vector ${\boldsymbol
\theta}$ corrupted by Gaussian noise with zero mean and unknown covariance
matrix ${\boldsymbol \Sigma}$. We propose minimax optimal methods of estimating
the $\ell_2$ norm of ${\boldsymbol \theta}$ and testing the hypothesis $H_0:
{\boldsymbol \theta}=0$ against sparse alternatives when only partial
information about ${\boldsymbol \Sigma}$ is available, such as an upper bound
on its Frobenius norm and the values of its diagonal entries to within an
unknown scaling factor. We show that the minimax rates of the estimation and
testing are leveraged not by the dimension of the problem but by the value of
the Frobenius norm of ${\boldsymbol \Sigma}$.
