  The existence of incompatible measurements, i.e. measurements which cannot be
performed simultaneously on a single copy of a quantum state, constitutes an
important distinction between quantum mechanics and classical theories. While
incompatibility might at first glance seem like an obstacle, it turns to be a
necessary ingredient to achieve the so-called quantum advantage in various
operational tasks like random access codes or key distribution. To improve our
understanding of how to quantify incompatibility of quantum measurements, we
define and explore a family of incompatibility measures based on
non-commutativity. We investigate some basic properties of these measures, we
show that they satisfy some natural information-processing requirements and we
fully characterize the pairs which achieve the highest incompatibility (in a
fixed dimension). We also consider the behavior of our measures under different
types of compositions. Finally, to link our new measures to existing results,
we relate them to a robustness-based incompatibility measure and two
operational scenarios: random access codes and entropic uncertainty relations.
