Reliable crystal structure prediction (CSP) at finite temperatures, including quantum anharmonic effects, remains challenging but is particularly important for systems containing light atoms such as superconducting hydrides. Here, we integrate machine-learned interatomic potentials (MLIPs) with the stochastic self-consistent harmonic approximation (SSCHA) to enable evolutionary CSP on the quantum anharmonic free-energy landscape. Using LaH10 as a test case, we compare three approaches: (i) active-learning MLIPs (AL-MLIPs) trained on the fly, (ii) universal MLIPs (uMLIPs), and (iii) temperature-dependent effective potentials (TDEPs) trained on SSCHA ensemble data. AL-MLIPs correctly predict the cubic Fm-3m phase but require thermodynamic perturbation theory corrections for consistent results. The foundation uMLIP Mattersim-5m enables SSCHA-based CSP without per-structure training, but fine-tuning is required for higher accuracy. Temperature-dependent effective potentials trained on SSCHA data enable accurate and very efficient CSP for large unit cells on the quantum anharmonic free-energy landscape. We demonstrate that quantum anharmonicity simplifies the free-energy landscape and is essential for the correct energy ranking of LaH10 structures.