  We show that there exists a positive real number $\delta>0$ such that for any
normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse
than canonical singularities, that is, the minimal log discrepancy of $X$ is
less than $1$, then the minimal log discrepancy of $X$ is not greater than
$1-\delta$. As applications, we show that the set of all non-canonical klt
Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all
klt Calabi-Yau $3$-folds are bounded from above.
