In summary, there is no known mechanism that can differentiate between two values of given spin, say up or down, and subsequently, "translate," that either value of spin into a corresponding direction in motion.
Separate question:
Were the time crystals developed in the 2010s looping in their quantum spin, or their linear momentum?
In 2016, https://en.wikipedia.org/w/index.php?title=Norman_Yao&action=edit&redlink=1 et al. proposed a different way to create discrete time crystals in spin systems. From there, Christopher Monroe and Mikhail Lukin independently confirmed this in their labs. Both experiments were published in Nature in 2017. In 2019 it was theoretically proven that a quantum time crystal can be realized in isolated systems with long-range multi-particle interactions (Wikipedia)
In these experiments, was it the quantum spin that underwent the looping time crystal phenomenon, or was it the actual linear motion which underwent this mechanism where it would repeat in time? If not, could we engineer a time crystal which loops over time not in linear motion, but with regards to its spin?