  Motivated by the recent discovery of near-room temperature superconductivity
in high-pressure superhydrides, we investigate from first-principles the
high-pressure superconducting phase diagram of the ternary Ca-B-H system, using
ab-initio evolutionary crystal structure prediction, and Density Functional
Perturbation Theory. We find that below 100 GPa all stable and weakly
metastable phases are insulating. This pressure marks the appearance of several
new chemically-forbidden phases on the hull of stability, and the first onset
of metalization in CaBH$_5$. Metallization is then gradually achieved at higher
pressure at different compositions. Among the metallic phases stable in the
Megabar regime, we predict two high-$T_c$ superconducting phases with CaBH$_6$
and Ca$_2$B$_2$H$_{13}$ compositions, with critical temperatures of 119 and 89
K at 300 GPa, respectively, surviving to lower pressures. Ternary hydrides will
most likely play a major role in superconductivity research in the coming
years; our study suggests that, in order to reduce the pressure for the onset
of metallicity and superconductivity, further explorations of ternary hydrides
should focus on elements less electronegative than boron.
