Complexity and instability of quantum motion near a quantum phase transition

We show that the number of harmonics of the Wigner function, recently proposed as a measure of quantum complexity, can be also used to characterize quantum phase transitions. The non-analytic behavior of this quantity in the neighborhood of a quantum phase transition is illustrated by means of the Dicke model and is compared to two well-known measures of the (in)stability of quantum motion, the quantum Loschmidt echo and the fidelity.