Classical matrix concentration inequalities are sharp up to a logarithmic factor. This logarithmic factor is necessary in the commutative case but unnecessary in many classical noncommutative cases. We will present some matrix concentration results that are sharp in many cases, where we overcome this logarithmic factor by using an easily computable quantity that captures noncommutativity. Joint work with Afonso Bandeira and Ramon van Handel. Paper: https://arxiv.org/abs/2108.06312