  In this work, global search for crystal structures of ternary Mg-Sc-H
hydrides (Mg$_x$Sc$_y$H$_z$) under high pressure ($100 \le P \le 200$ GPa) were
performed using the evolutionary algorithm and first-principles calculations.
Based on them, we computed the thermodynamic convex hull and pressure-dependent
phase diagram of Mg$_x$Sc$_y$H$_z$ for $z/(x+y) < 4$. We have identified the
stable crystal structures of four thermodynamically stable compounds with the
higher hydrogen content, i.e., $R\bar{3}m$-MgScH$_{6}$,
$C2/m$-Mg$_{2}$ScH$_{10}$, $Immm$-MgSc$_{2}$H$_{9}$ and
$Pm\bar{3}m$-Mg(ScH$_{4}$)$_{3}$. Their superconducting transition temperatures
were computationally predicted by the McMillan-Allen-Dynes formula combined
with first-principles phonon calculations. They were found to exhibit
superconductivity; among them, $R\bar{3}m$-MgScH$_{6}$ was predicted to have
the highest $T_{c}$ (i.e. 23.34 K) at 200 GPa.
