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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.