  Combining robust magnetism, strong spin-orbit coupling and unique
thickness-dependent properties of van der Waals crystals could enable new
spintronics applications. Here, using density functional theory, we propose the
(MnSb$_2$Te$_4$)$\cdot$(Sb$_2$Te$_3$)$_n$ family of stoichiometric van der
Waals compounds that harbour multiple topologically-nontrivial magnetic phases.
In the groundstate, the first three members of the family, i.e. MnSb$_2$Te$_4$,
($n=0$), MnSb$_4$Te$_7$, ($n=1$), and MnSb$_6$Te$_{10}$, ($n=2$), are 3D
antiferromagnetic topological insulators (AFMTIs), while for $n \geq 3$ a
special phase is formed, in which a nontrivial topological order coexists with
a partial magnetic disorder in the system of the decoupled 2D ferromagnets,
whose magnetizations point randomly along the third direction. Furthermore, due
to a weak interlayer exchange coupling, these materials can be field-driven
into the FM Weyl semimetal ($n=0$) or FM axion insulator states ($n \geq 1$).
Finally, in two dimensions we reveal these systems to show intrinsic quantum
anomalous Hall and AFM axion insulator states, as well as quantum Hall state,
achieved under external magnetic field, but without Landau levels. Our results
provide a solid computational proof that MnSb$_2$Te$_4$, is not topologically
trivial as was previously believed that opens possibilities of realization of a
wealth of topologically-nontrivial states in the
(MnSb$_2$Te$_4$)$\cdot$(Sb$_2$Te$_3$)$_n$ family.
