  In this paper we exploit the utility of the triangle symbol which has a
complicated expression in terms of spider diagrams in ZX-calculus, and its role
within the ZX-representation of AND-gates in particular. First, we derive
spider nest identities which are of key importance to recent developments in
quantum circuit optimisation and T-count reduction in particular. Then, using
the same rule set, we prove a completeness theorem for quantum Boolean circuits
(QBCs) whose rewriting rules can be directly used for a new method of T-count
reduction. We give an algorithm based on this method and show that the results
of our algorithm outperform the results of all the previous best
non-probabilistic algorithms.
