Abstract
Reference frames are used to parameterize measurements of physical effects, but since their practical realization uses material objects, they may affect observations performed in a combined quantum state of the measured system together with the frame. Here, a procedure is used that makes it possible to describe non-monotonic reference scales in a quantum treatment, revealing large quantum effects in the measured system whenever a reference frame encounters a turning point. Subtle quantum correlations in the combined state of system and frame, and more broadly the concept of relational quantum mechanics, can be tested via a characteristic and surprisingly large shift in the measured value.