  We establish a kind of subadjunction formula for quasi-log canonical pairs.
As an application, we prove that a connected projective quasi-log canonical
pair whose quasi-log canonical class is anti-ample is simply connected and
rationally chain connected. We also supplement the cone theorem for quasi-log
canonical pairs. More precisely, we prove that every negative extremal ray is
spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity
for quasi-log canonical pairs.
