  Motivated by the recent experiment on $\rm{K_2Cu_3O\left(SO_4\right)_3}$, an
edge-shared tetrahedral spin-cluster compound [M. Fujihala \textit{et al.},
Phys. Rev. Lett. \textbf{120}, 077201 (2018)], we investigate two-leg
spin-cluster ladders with the plaquette number $n_p$ in each cluster up to six
by the density-matrix renormalization group method. We find that the phase
diagram of such ladders strongly depends on the parity of $n_p$. For even
$n_p$, the phase diagram has two phases, one is the Haldane phase, and the
other is the cluster rung-singlet phase. For odd $n_p$, there are four phases,
which are a cluster-singlet phase, a cluster rung-singlet phase, a Haldane
phase and an even Haldane phase. Moreover, in the latter case the region of the
Haldane phase increases while the cluster-singlet phase and the even Haldane
phase shrink as $n_p$ increases. We thus conjecture that in the large $n_p$
limit, the phase diagram will become independent of $n_p$. By analysing the
ground-state energy and entanglement entropy we obtain the order of the phase
transtions. In particular, for $n_p=1$ there is no phase transition between the
even Haldane phase and the cluster-singlet phase while for other odd $n_p$
there is a first-order phase transition. Our work provides comprehensive phase
diagrams for these cluster-based models and may be helpful to understand
experiments on related materials.
