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As spectroscopic surveys continue to grow in size, the problem of classifying spectra targeted as quasars (QSOs) will need to move beyond its historical reliance on human experts. Instead, automatic classifiers will increasingly become the dominant classification method, leaving only small fractions of spectra to be visually inspected in ambiguous cases. In order to maximise classification accuracy, making best use of available classifiers will be of great importance, particularly when looking to identify and eliminate distinctive failure modes. In this work, we demonstrate that the machine learning-based classifier QuasarNET will be of use for future surveys such as the Dark Energy Spectroscopic Instrument (DESI), comparing its performance to the DESI pipeline classifier redrock. During the first of four passes across its footprint DESI will need to select high-$z$ ($z\geq2.1$) QSOs for reobservation, and so we first assess the classifiers' performance at identifying high-$z$ QSOs from single-exposure spectra. We then quantify the classifiers' abilities to construct QSO catalogues in both low- and high-$z$ bins, using coadded spectra to simulate end-of-survey data. For such tasks, QuasarNET is able to outperform redrock in its current form, identifying approximately 99% of high-$z$ QSOs from single exposures and producing QSO catalogues with sub-percent levels of contamination. By combining QuasarNET and redrock's outputs, we can further improve the classification strategies to identify up to 99.5% of high-$z$ QSOs from single exposures and reduce final QSO catalogue contamination to below 0.5%. These combined strategies address DESI's QSO classification needs effectively.