  We have systematically calculated the mass spectra for S-wave and P-wave
fully-charm $c\bar{c}c\bar{c}$ and fully-bottom $b\bar{b}b\bar{b}$ tetraquark
states in the $\mathbf{8}_{[Q\bar{Q}]}\otimes \mathbf{8}_{[Q\bar{Q}]}$ color
configuration, by using the moment QCD sum rule method. The masses for the
fully-charm $c\bar cc\bar c$ tetraquark states are predicted about $6.3-6.5$
GeV for S-wave channels and $7.0-7.2$ GeV for P-wave channels. These results
suggest the possibility that there are some $\mathbf{8}_{[c\bar{c}]}\otimes
\mathbf{8}_{[c\bar{c}]}$ components in LHCb's di-$J/\psi$ structures. For the
fully-bottom $b\bar{b}b\bar{b}$ system, their masses are calculated around 18.2
GeV for S-wave tetraquark states while 18.4-18.8 GeV for P-wave ones, which are
below the $\eta_b\eta_b$ and $\Upsilon(1S)\Upsilon(1S)$ two-meson decay
thresholds.
