  We give a simple proof of the matrix Spencer conjecture up to
poly-logarithmic rank: given symmetric $d \times d$ matrices $A_1,\ldots,A_n$
each with $\|A_i\|_{\mathsf{op}} \leq 1$ and rank at most $n/\log^3 n$, one can
efficiently find $\pm 1$ signs $x_1,\ldots,x_n$ such that their signed sum has
spectral norm $\|\sum_{i=1}^n x_i A_i\|_{\mathsf{op}} = O(\sqrt{n})$. This
result also implies a $\log n - \Omega( \log \log n)$ qubit lower bound for
quantum random access codes encoding $n$ classical bits with advantage $\gg
1/\sqrt{n}$.
  Our proof uses the recent refinement of the non-commutative Khintchine
inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices
with correlated Gaussian entries.
