  Crystalline $\rm Bi_2Se_3$ is one of the most explored three-dimensional
topological insulator, with a $0.3\;\rm eV$ energy gap making it promising for
applications. Its amorphous counterpart could bring to light new possibilities
for large scale synthesis and applications. Using ab initio molecular dynamics
simulations, we have studied realistic amorphous $\rm Bi_2Se_3$ phases
generated by different processes of melting, quenching, and annealing.
Extensive structural and electronic characterizations show that the melting
process induces an energy gap decrease ruled by growth of the defective local
environments. This behavior dictates a weak stability of the topological phase
to disorder, characterized by the spin Bott index. Interestingly, we identify
the occurrence of topologically trivial surface states in amorphous $\rm
Bi_2Se_3$ that show a strong resemblance with standard helical topological
states. Our results and methods advance the search of topological phases in
three-dimensional amorphous solids.
