We report the discovery by the HATSouth project of 5 new transiting hot Jupiters (HATS-54b through HATS-58Ab). HATS-54b, HATS-55b and HATS-58Ab are prototypical short period ($P = 2.5-4.2$ days, $R_psim1.1-1.2$ $R_J$) hot-Jupiters that span effective temperatures from 1350 K to 1750 K, putting them in the proposed region of maximum radius inflation efficiency. The HATS-58 system is composed of two stars, HATS-58A and HATS-58B, which are detected thanks to Gaia DR2 data and which we account for in the joint modelling of the available data --- with this, we are led to conclude that the hot jupiter orbits the brighter HATS-58A star. HATS-57b is a short-period (2.35-day) massive (3.15 $M_J$) 1.14 $R_J$, dense ($2.65pm0.21$ g cm$^{-3}$) hot-Jupiter, orbiting a very active star ($2%$ peak-to-peak flux variability). Finally, HATS-56b is a short period (4.32-day) highly inflated hot-Jupiter (1.7 $R_J$, 0.6 $M_J$), which is an excellent target for future atmospheric follow-up, especially considering the relatively bright nature ($V=11.6$) of its F dwarf host star. This latter exoplanet has another very interesting feature: the radial velocities show a significant quadratic trend. If we interpret this quadratic trend as arising from the pull of an additional planet in the system, we obtain a period of $P_c = 815^{+253}_{-143}$ days for the possible planet HATS-56c, and a minimum mass of $M_csin i_c = 5.11 pm 0.94$ $M_J$. The candidate planet HATS-56c would have a zero-albedo equilibrium temperature of $T_textrm{eq}=332pm 50$ K, and thus would be orbiting close to the habitable zone of HATS-56. Further radial-velocity follow-up, especially over the next two years, is needed to confirm the nature of HATS-56c.
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