Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d × d matrices A1,…,An each with ||Ai||op ≤ 1 and rank at most n/log3 n, one can efficiently find ± 1 signs x1,…,xn such that their signed sum has spectral norm ||∑i=1n xi Ai||op = O(√n). This result also implies a logn − Ω( loglogn) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫ 1/√n.