Motivated by the recent synthesis of two insulating Li$_2$IrO$_3$ polymorphs, where Ir$^{4+}$ $S_{eff}$=1/2 moments form 3D (harmonic) honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $beta$-Li$_2$IrO$_3$ hyperhoneycomb lattice and $gamma$-Li$_2$IrO$_3$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal 90$^circ$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with emergent Majorana fermions and Z$_2$ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with $T_c approx |K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $ell$ is relatively large in these structures, we construct an $ellrightarrow infty$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.
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