  Quantum measurements and phase transitions are seemingly uncorrelated topics,
but here we show that phase transitions occur in sequential quantum
measurements. We find that the probability distribution of the measurement
results of a sequence of quantum measurements on a two-level system (e.g. a
qubit) is equivalent to the Boltzmann distribution of a classical lattice spin
model. So the measurement results present phase transitions similar to those in
the lattice spin model. In sequential commuting positive-operator valued
measurements, the probability distribution is mapped to a long-range Ising
model in the weak-measurement regime, and a projective measurement emerges from
a sequence of weak measurement when the strength or the number of measurements
becomes above certain critical values, which correspond to a second-order
ferromagnetic phase transition of the lattice spin model. These findings not
only provide new insights on sequential quantum measurements, but may also have
potential applications in quantum technologies.
