Threshold access structures of secret sharing schemes capture a scenario in which all the participants have the same weight (or power) and their contributions are equal. However, in some situations such as gradation among officials in an organization, the participants have different weights. Hierarchical access structures capture those natural scenarios, where different levels of hierarchy are present and a participant belongs precisely to one of them. Although an extensive research addressing the issues of cheater identifiability and robustness have been done for threshold secret sharing, no such research has been carried out for hierarchical secret sharing (HSS). This paper resolves this long-standing open issue by presenting definitions and constructions of both cheater identifiable and robust HSS schemes secure against rushing adversary, in the information-theoretic setting.