  Topological frustration (or topological mechanics) is the existence of
classical zero modes that are robust to many but not all distortions of the
Hamiltonian. It arises naturally from locality in systems whose interactions
form a set of constraints such as in geometrically frustrated magnets and balls
and springs metamaterials. For a magnet whose classical limit exhibits
topological frustration, an important question is what happens to this topology
when the degrees of freedom are quantized and whether such frustration could
lead to exotic quantum phases of matter like a spin liquid. We answer these
questions for a geometrically frustrated spin ladder model. It has the feature
of having infinitely many conserved quantities that aid the solution. We find
classical zero modes all get lifted by quantum fluctuations and the system is
left with a unique rung singlet ground state -- a trivial quantum spin liquid.
Moreover, we find low-energy eigenstates corresponding to known symmetry
protected topological (SPT) ground states, and a special role of $SU(2)$
symmetry, that it demands the existence of extra dimensions of classical zero
modes -- the phenomena we call symmetry-enriched topological frustration
(SETF). These results suggest small violations of the conservation laws in the
nearly SETF regime could lead to quantum scars. We further study a
two-dimensional bilayer triangular lattice model and find a similar SETF
phenomena which also leads to suppressed low-energy topological eigenstates in
the quantum regime. These results suggest that in the absence of magnetic
order, classical topological frustration manifests at finite spin as
asymptotically low energy modes with support for exotic quantum phenomena.
