  Given vectors $\mathbb{v}_1, \ldots, \mathbb{v}_n \in \mathbb{R}^d$ with
Euclidean norm at most $1$ and $\mathbb{x}_0 \in [-1,1]^n$, our goal is to
sample a random signing $\mathbb{x} \in \{\pm 1\}^n$ with
$\mathbb{E}[\mathbb{x}] = \mathbb{x}_0$ such that the operator norm of the
covariance of the signed sum of the vectors $\sum_{i=1}^n \mathbb{x}(i)
\mathbb{v}_i$ is as small as possible. This problem arises from the algorithmic
discrepancy theory and its application in the design of randomized experiments.
It is known that one can sample a random signing with expectation
$\mathbb{x}_0$ and the covariance operator norm at most $1$.
  In this paper, we prove two hardness results for this problem. First, we show
it is NP-hard to distinguish a list of vectors for which there exists a random
signing with expectation ${\bf 0}$ such that the operator norm is $0$ from
those for which any signing with expectation ${\bf 0}$ must have the operator
norm $\Omega(1)$. Second, we consider $\mathbb{x}_0 \in [-1,1]^n$ whose entries
are all around an arbitrarily fixed $p \in [-1,1]$. We show it is NP-hard to
distinguish a list of vectors for which there exists a random signing with
expectation $\mathbb{x}_0$ such that the operator norm is $0$ from those for
which any signing with expectation ${\bf 0}$ must have the operator norm
$\Omega((1-|p|)^2)$.
