  The problem of minimizing the maximum of $N$ convex, Lipschitz functions
plays significant roles in optimization and machine learning. It has a series
of results, with the most recent one requiring $O(N\epsilon^{-2/3} +
\epsilon^{-8/3})$ queries to a first-order oracle to compute an
$\epsilon$-suboptimal point. On the other hand, quantum algorithms for
optimization are rapidly advancing with speedups shown on many important
optimization problems. In this paper, we conduct a systematic study for quantum
algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz
functions. On one hand, we develop quantum algorithms with an improved
complexity bound of $\tilde{O}(\sqrt{N}\epsilon^{-5/3} + \epsilon^{-8/3})$. On
the other hand, we prove that quantum algorithms must take
$\tilde{\Omega}(\sqrt{N}\epsilon^{-2/3})$ queries to a first order quantum
oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic
factors.
