  We develop a moment QCD sum rule method augmented by fundamental inequalities
to study the existence of exotic doubly hidden-charm/bottom tetraquark states
made of four heavy quarks. Using the compact diquark-antidiquark configuration,
we calculate the mass spectra of these tetraquark states. There are 18
hidden-charm $cc\bar c\bar c$ tetraquark currents with $J^{PC} = 0^{++}$,
$0^{-+}$, $0^{--}$, $1^{++}$, $1^{+-}$, $1^{-+}$, $1^{--}$, and $2^{++}$. We
use them to perform QCD sum rule analyses, and the obtained masses are all
higher than the spontaneous dissociation thresholds of two charmonium mesons,
which are thus their dominant decay modes. The masses of the corresponding
hidden-bottom $bb\bar b\bar b$ tetraquarks are all below or very close to the
thresholds of the $\Upsilon(1S)\Upsilon(1S)$ and $\eta_b(1S)\eta_b(1S)$, except
one current of $J^{PC}=0^{++}$. Hence, we suggest to search for the doubly
hidden-charm states in the $J/\psi J/\psi$ and $\eta_c(1S)\eta_c(1S)$ channels.
