Not yet reviewed by Pith; the record is open.

The low efficiency of conventional liquefaction technologies based on the Joule-Thomson expansion makes liquid hydrogen currently not attractive enough for large-scale energy-related technologies that are important for the transition to a carbon-neutral society. Magnetocaloric hydrogen liquefaction has great potential to achieve higher efficiency and is therefore a crucial enabler for affordable liquid hydrogen. Cost-effective magnetocaloric materials with large magnetic entropy and adiabatic temperature changes in the temperature range of 77 $\sim$ 20 K under commercially practicable magnetic fields are the foundation for the success of magnetocaloric hydrogen liquefaction. Heavy rare-earth-based magnetocaloric intermetallic compounds generally show excellent magnetocaloric performances, but the heavy rare-earth elements (Gd, Tb, Dy, Ho, Er, and Tm) are highly critical in resources. Yttrium and light rare-earth elements (La, Ce, Pr, and Nd) are relatively abundant, but their alloys generally show less excellent magnetocaloric properties. A dilemma appears: higher performance or lower criticality? In this review, we study how cryogenic temperature influences magnetocaloric performance by first reviewing heavy rare-earth-based intermetallic compounds. Next, we look at light rare-earth-based, "mixed" rare-earth-based, and Gd-based intermetallic compounds with the nature of the phase transition order taken into consideration, and summarize ways to resolve the dilemma.