  Quantum resource theory under different classes of quantum operations
advances multiperspective understandings of inherent quantum-mechanical
properties, such as quantum coherence and quantum entanglement. We establish
hierarchies of different operations for manipulating coherence and entanglement
in distributed settings, where at least one of the two spatially separated
parties are restricted from generating coherence. In these settings, we
introduce new classes of operations and also characterize those maximal, i.e.,
the resource-non-generating operations, progressing beyond existing studies on
incoherent versions of local operations and classical communication and those
of separable operations. The maximal operations admit a
semidefinite-programming formulation useful for numerical algorithms, whereas
the existing operations not. To establish the hierarchies, we prove a sequence
of inclusion relations among the operations by clarifying tasks where
separation of the operations appears. We also demonstrate an asymptotically
non-surviving separation of the operations in the hierarchy in terms of
performance of the task of assisted coherence distillation, where a separation
in a one-shot scenario vanishes in the asymptotic limit. Our results serve as
fundamental analytical and numerical tools to investigate interplay between
coherence and entanglement under different operations in the resource theory.
