  Topological orders have been intrinsically identified in a class of systems
such as fractional quantum Hall states and spin liquids. Accessing such states
often requires extreme conditions such as low temperatures, high magnetic
fields, pure samples, etc. Another approach would be to engineer the
topological orders in systems with more accessible ingredients. In this work,
we present networks of Majorana bound states, which are currently accessible in
semiconductor nanowires proximitized to conventional superconductors, and show
that the effective low-energy theory is topologically ordered. We first
demonstrate the main principles in a lattice made of Kitaev superconducting
chains comprising both spin species. The lattice is coupled to free magnetic
moments through the Kondo interaction. We then show that at the weak coupling
limit, effective ring spin interactions are induced between magnetic moments
with a topological order enjoying a local $\mathbb{Z}_2 \times \mathbb{Z}_2$
gauge symmetry. We then show that the same topological order and also the
$\mathbb{Z}_2$ one can be engineered in architecture patterns of semiconductor
nanowires hosting Majorana bound states. The basic blocks of patterns are the
time-reversal Majorana Cooper boxes coupled to each other by metallic leads,
and the Majorana states are allowed to tunnel to quantum dots sitting on the
vertices of the lattices. In the limit of strong onsite Coulomb interactions,
where the charge fluctuations are suppressed, the magnetic moments of dots on
the square and honeycomb lattices are described by topologically ordered spin
models with underlying $\mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$
gauge symmetries, respectively. Finally, we show that the latter topological
order can also be realized in a network of purely Majorana zero modes in the
absence of coupling to quantum dots.
