We perform a systematic study of the five known QPE host galaxies with the aim of providing new insights into their nature. We analyse new and archival medium resolution optical spectroscopy of the QPE hosts. We measure emission (and absorption) line fluxes, their ratios and equivalent widths (EWs), to locate the QPE hosts on diagnostic diagrams. We also measure the velocity dispersion of the stellar absorption lines to estimate their black hole masses. All QPE host galaxies show emission lines in their optical spectra. Based on their ratios and EWs, we find evidence for the presence of an active galactic nucleus in all sources, including those previously reported as passive. We measure velocity dispersions between 36 and 90km/s, implying the presence of low mass (10^5-6.7^ solar masses) black holes, consistent with literature findings. Finally, we find a significant over-representation (2/5 sources, or a factor of 13^+13^_-10_ quiescent Balmer strong (post-starburst) galaxies among QPE hosts.