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We study the mass spectra for the cccc and bbbb tetraquark states by developing a moment sum rule method. Our results show that the bbbb tetraquarks lie below the threshold of ηb(1S) ηb(1S ). They are probably stable and very narrow. The masses for the doubly hidden-charm states cccc are higher than the spontaneous dissociation thresholds of two charmonium mesons. We suggest to search for such states in the J/Ψ/J/Ψ and Ψc(1S) Ψc(1S) channels.