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Measurements allow efficient preparation of interesting quantum many-bodystates with long-range entanglement, conditioned on additional transformationsbased on measurement outcomes. Here, we demonstrate that the so-calledconformal quantum critical points (CQCP) can be obtained by performing generalsingle-site measurements in an appropriate basis on the cluster states in$d\geq2$. The equal-time correlators of the said states are described bycorrelation functions of certain $d$-dimensional classical models at finitetemperatures and feature spatial conformal invariance. This establishes anexact correspondence between the measurement-prepared critical states andconformal field theories of a range of critical spin models, including familiarIsing models and gauge theories. Furthermore, by mapping the long-rangeentanglement structure of measured quantum states into the correlations of thecorresponding thermal spin model, we rigorously establish the stabilitycondition of the long-range entanglement in the measurement-prepared quantumstates deviating from the ideal setting. Most importantly, we describeprotocols to decode the resulting quantum phases and transitions withoutpost-selection, thus transferring the exponential measurement complexity to apolynomial classical computation. Therefore, our findings suggest a novelmechanism in which a quantum critical wavefunction emerges, providing newpractical ways to study quantum phases and conformal quantum critical points.

Decoding Measurement-Prepared Quantum Phases and Transitions: from Ising model to gauge theory, and beyond