  For a pair of observables, they are called "incompatible", if and only if the
commutator between them does not vanish, which represents one of the key
features in quantum mechanics. The question is, how can we characterize the
incompatibility among three or more observables? Here we explore one possible
route towards this goal through Heisenberg's uncertainty relations, which
impose fundamental constraints on the measurement precisions for incompatible
observables. Specifically, we quantify the incompatibility by the optimal
state-independent bounds of additive variance-based uncertainty relations. In
this way, the degree of incompatibility becomes an intrinsic property among the
operators, but not on the quantum state. To justify our case, we focus on the
incompatibility of spin systems. For an arbitrary setting of two or three
linearly-independent Pauli-spin operators, the incompatibility is analytically
solved, the spins are maximally incompatible if and only if they are orthogonal
to each other. On the other hand, the measure of incompatibility represents a
versatile tool for applications such as testing entanglement of bipartite
states, and EPR-steering criteria.
