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We seek to develop better upper bound guarantees on the depth of quantum <inline-formula><tex-math notation="LaTeX">$\text {CZ}$</tex-math></inline-formula> gate, <sc>cnot</sc> gate, and Clifford circuits than those reported previously. We focus on the number of qubits <inline-formula><tex-math notation="LaTeX">$n\,{\leq }\,$</tex-math></inline-formula>1 345 000 (de Brugi&#x00E8;re <italic>et al.</italic>, 2021), which represents the most practical use case. Our upper bound on the depth of <inline-formula><tex-math notation="LaTeX">$\text {CZ}$</tex-math></inline-formula> circuits is <inline-formula><tex-math notation="LaTeX">$\lfloor n/2 + 0.4993{\cdot }\log ^{2}(n) + 3.0191{\cdot }\log (n) - 10.9139\rfloor$</tex-math></inline-formula>, improving the best-known depth by a factor of roughly 2. We extend the constructions used to prove this upper bound to obtain depth upper bound of <inline-formula><tex-math notation="LaTeX">$\lfloor n + 1.9496{\cdot }\log ^{2}(n) + 3.5075{\cdot }\log (n) - 23.4269 \rfloor$</tex-math></inline-formula> for <sc>cnot</sc> gate circuits, offering an improvement by a factor of roughly <inline-formula><tex-math notation="LaTeX">$4/3$</tex-math></inline-formula> over the state of the art, and depth upper bound of <inline-formula><tex-math notation="LaTeX">$\lfloor 2n + 2.9487{\cdot }\log ^{2}(n) + 8.4909{\cdot }\log (n) - 44.4798\rfloor$</tex-math></inline-formula> for Clifford circuits, offering an improvement by a factor of roughly <inline-formula><tex-math notation="LaTeX">$5/3$</tex-math></inline-formula>.