We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to push forward some known constructs involving relativized unambiguity based complexity classes (UP and Promise- UP) to new constructs involving arbitrary levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical assembly of
nondeterministic polynomial-time Turing machines, and so our techniques differ substantially, in applicability and in nature, from standard techniques (such as the switching lemma [Hås87]), which are known to play roles in carrying out similar generalizations.
Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč [CHV93] on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP.