  In present work, spectrum of the $S$-wave fully-heavy tetraquark states
$QQ\bar{Q}\bar{Q}$ ($Q=c,b$), i.e., $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$,
$cc\bar{b}\bar{b}$/$bb\bar{c}\bar{c}$, $bc\bar{c}\bar{c}$/ $cc\bar{b}\bar{c}$,
$bb\bar{c}\bar{b}$/$cb\bar{b}\bar{b}$, and $bc\bar{b}\bar{c}$ are
systematically investigated within an nonrelativistic constituent quark model,
in which the Instanton-induced and one-gluon-exchange interactions are taken
into account as the residual spin-dependent hyperfine interaction. Our results
show that the states with $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ components
could be located around $ 6500$ MeV and $ 19200$ MeV, respectively. Based on
our calculations, the new $X(6900)$ state observed by LHCb may be not a ground
$cc\bar{c}\bar{c}$ tetraquark state, while it could be an orbitally or radially
excited state of $cc\bar{c}\bar{c}$ system. On the other hand, the recently
reported $X(6600)$ state by CMS and ATLAS can be explained as a ground
$cc\bar{c}\bar{c}$ tetraquark state with spin-parity $J^{PC} =0^{++}$.
