We have systematically calculated the mass spectra for S-wave and P-wave fully-charm $cbar{c}cbar{c}$ and fully-bottom $bbar{b}bbar{b}$ tetraquark states in the $mathbf{8}_{[Qbar{Q}]}otimes mathbf{8}_{[Qbar{Q}]}$ color configuration, by using the moment QCD sum rule method. The masses for the fully-charm $cbar ccbar c$ 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 $mathbf{8}_{[cbar{c}]}otimes mathbf{8}_{[cbar{c}]}$ components in LHCbs di-$J/psi$ structures. For the fully-bottom $bbar{b}bbar{b}$ system, their masses are calculated around 18.2 GeV for S-wave tetraquark states while 18.4-18.6 GeV for P-wave ones, which are below the $eta_beta_b$ and $Upsilon(1S)Upsilon(1S)$ two-meson decay thresholds.
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