  Motivated by the observation of exotic structure around 6900MeV in the
$J/\psi$-pair mass spectrum using proton-proton collision data by the LHCb
collaboration, we study the spectra of fully-heavy tetraquarks within
Bethe-Salpeter equation and Regge trajectory relation. The $X(6900)$ may be
explained as a radially excited state with quark content $cc\bar{c}\bar{c}$ and
spin-parity $0^{++}(3S)$ or $2^{++}(3S)$ or an orbitally excited $2P$ state.
New $cc\bar{c}\bar{c}$ structures around 6.0GeV, 6.5GeV, and 7.1GeV are
predicted together. Other $bb\bar{b}\bar{b}$ and $bc\bar{b}\bar{c}$ structures
which may be experimentally prominent are discussed. On the other hand, the
fully-heavy S-wave tetraquark productions at hadron colliders are investigated
and their cross-sections are obtained.
