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We propose a new dynamical method to connect equilibrium quantum phasetransitions and quantum coherence using out-of-time-order correlations (OTOCs).Adopting the iconic Lipkin-Meshkov-Glick and transverse-field Ising models asillustrative examples, we show that an abrupt change in coherence andentanglement of the ground state across a quantum phase transition isobservable in the spectrum of multiple quantum coherence (MQC) intensities,which are a special type of OTOC. We also develop a robust protocol to obtainthe relevant OTOCs using quasi-adiabatic quenches through the ground statephase diagram. Our scheme allows for the detection of OTOCs withouttime-reversal of coherent dynamics, making it applicable and important for abroad range of current experiments where time-reversal cannot be achieved byinverting the sign of the underlying Hamiltonian.

Detecting out-of-time-order correlations via quasi-adiabatic echoes as a tool to reveal quantum coherence in equilibrium quantum phase transitions