In this work we study the mass spectra of the fully-heavy tetraquark systems, i.e. $ccbar{c}bar{c}$, $bbbar{b}bar{b}$, $bbbar{c}bar{c}/ccbar{b}bar{b}$, $bcbar{c}bar{c}/ccbar{b}bar{c}$, $bcbar{b}bar{b}/bbbar{b}bar{c}$, and $bcbar{b}bar{c}$, within a potential model by including the linear confining potential, Coulomb potential, and spin-spin interactions. It shows that the linear confining potential has important contributions to the masses and is crucial for our understanding of the mass spectra of the fully-heavy tetraquark systems. For the fully-heavy tetraquarks $Q_1Q_2bar{Q}_3bar{Q}_4$ our explicit calculations suggest that no bound states can be formed below the thresholds of any meson pairs $(Q_1bar{Q}_3)$-$(Q_2bar{Q}_4)$ or $(Q_1bar{Q}_4)$-$(Q_2bar{Q}_3)$. Thus, we do not expect narrow fully-heavy tetraquark states to be existing in experiments.
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