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The spectral features of water clusters provide important information on their structure and dynamics and can assist in deciphering the nature of the local environment of aqueous solutions in a variety of different conditions. Accurately capturing these features via numerical simulations is a nontrivial task that typically requires a sophisticated combination of high-level electronic structure methods and costly quantum dynamics techniques. We present results of molecular dynamics simulations of the IR spectra of protonated water clusters, ranging from the monomer to the tetramer, obtained via the combination of highly accurate machine-learned potential energy surfaces (PES) and dipole moment surfaces (DMS), and the quantum thermal bath (QTB) methodology which facilitates cost-effective inclusion of NQEs in molecular dynamics simulations. We compare our results with previous theoretical and experimental studies and show that this combination provides a significantly cheaper, yet still suitably accurate, alternative to approaches such as ring polymer molecular dynamics and its thermostated variant (TRPMD), often used for these calculations.