  Background: Zr region is characterized by very rapid changes in the ground
state structure of the nuclei. In particular, the onset of deformation when
passing from $^{98}$Zr to $^{100}$Zr is one of the fastest ever observed in the
nuclear chart. It has been probed both experimental and theoretically that
certain low-lying excited states of Zr isotopes own different shapes than the
ground state.
  Purpose: We intend to disentangle the interplay between the sudden changes in
the ground state shape, i.e., the existence of a quantum phase transition, and
the presence in the spectra of coexisting states with very different
deformation, i.e., the presence of shape coexistence.
  Method: We rely on a previous calculation using the Interacting Boson Model
with Configuration Mixing (IBM-CM) which reproduces in detail the spectroscopic
properties of $^{96-110}$Zr. This IBM-CM calculation allows to compute
mean-field energy surfaces, wave functions and any other observable related
with the presence of shape coexistence or with a quantum phase transition.
  Results: We obtain energy surfaces and the equilibrium value of the
deformation parameter $\beta$, the U(5) decomposition of the wave functions and
the density of states.
  Conclusions: We confirm that Zr is a clear example of quantum phase
transition that originates from the crossing of two configurations with a very
different degree of deformation. Moreover, we observe how the intruder
configuration exhibits its own evolution which resembles a quantum phase
transition too.
