  We introduce quantum hypercube states, a class of continuous-variable quantum
states that are generated as orthographic projections of hypercubes onto the
quadrature phase-space of a bosonic mode. In addition to their interesting
geometry, hypercube states display phase-space features much smaller than
Planck's constant, and a large volume of Wigner-negativity. We theoretically
show that these features make hypercube states sensitive to displacements at
extremely small scales in a way that is surprisingly robust to initial thermal
occupation and to small separation of the superposed state-components. In a
high-temperature proof-of-principle optomechanics experiment we observe, and
match to theory, the signature outer-edge vertex structure of hypercube states.
