  Long-range entanglement--the backbone of topologically ordered states--cannot
be created in finite time using local unitary circuits, or equivalently,
adiabatic state preparation. Recently it has come to light that single-site
measurements provide a loophole, allowing for finite-time state preparation in
certain cases. Here we show how this observation imposes a complexity hierarchy
on long-range entangled states based on the minimal number of measurement
layers required to create the state, which we call "shots". First, similar to
Abelian stabilizer states, we construct single-shot protocols for creating any
non-Abelian quantum double of a group with nilpotency class two (such as $D_4$
or $Q_8$). We show that after the measurement, the wavefunction always
collapses into the desired non-Abelian topological order, conditional on
recording the measurement outcome. Moreover, the clean quantum double ground
state can be deterministically prepared via feedforward--gates which depend on
the measurement outcomes. Second, we provide the first constructive proof that
a finite number of shots can implement the Kramers-Wannier duality
transformation (i.e., the gauging map) for any solvable symmetry group. As a
special case, this gives an explicit protocol to prepare twisted quantum double
for all solvable groups. Third, we argue that certain topological orders, such
as non-solvable quantum doubles or Fibonacci anyons, define non-trivial phases
of matter under the equivalence class of finite-depth unitaries and
measurement, which cannot be prepared by any finite number of shots. Moreover,
we explore the consequences of allowing gates to have exponentially small
tails, which enables, for example, the preparation of any Abelian anyon theory,
including chiral ones. This hierarchy paints a new picture of the landscape of
long-range entangled states, with practical implications for quantum
simulators.
