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We present a first-principles study of the electronic and magnetic properties of epitaxial interfaces between the Heusler compounds Ti$_2$MnIn and Ni$_2$MnIn and the III-V semiconductors, InSb and InAs, respectively. We use density functional theory (DFT) with a machine-learned Hubbard $U$ correction determined by Bayesian optimization. We evaluate these interfaces for prospective applications in Majorana-based quantum computing and spintronics. In both interfaces, states from the Heusler penetrate into the gap of the semiconductor, decaying within a few atomic layers. The magnetic interactions at the interface are weak and local in space and energy. Magnetic moments of less than 0.1 $μ_B$ are induced in the two atomic layers closest to the interface. The induced spin polarization around the Fermi level of the semiconductor also decays within a few atomic layers. The decisive factor for the induced spin polarization around the Fermi level of the semiconductor is the spin polarization around the Fermi level in the Heusler, rather than the overall magnetic moment. As a result, the ferrimagnetic narrow-gap semiconductor Ti$_2$MnIn induces a more significant spin polarization in the InSb than the ferromagnetic metal Ni$_2$MnIn induces in the InAs. This is explained by the position of the transition metal $d$ states in the Heusler with respect to the Fermi level. Based on our results, these interfaces are unlikely to be useful for Majorana devices but could be of interest for spintronics.