A previously established information theoretic approach for dynamical hierarchies is hereby revisited for the case when a higher hierarchical level, which is observable in the realm of classical physics, arises from quantum dynamics. The focus of the paper is on the description of information transfer from quantum to classical level. As an example of complex functionality emerging at a higher level from quantum dynamics in a biological system, a case study of coherent spin dynamics, which is hypothesised to be one of the mechanisms of avian magnetoreception, is presented within such an approach. The findings may be useful for better understanding of the phenomena emergent from quantum processes which occur in biological complex systems. Moreover, the lessons learned may provide valuable knowledge for nature-inspired engineering in a bottom-up fashion within the field of nanotechnology.