We study phase transitions and algorithms for refuting CSPs arising in hierarchical clustering (as well as ranking, and ordinary CSPs). Here, $n$ variables are assigned to leaves of a tree, so as to satisfy $m$ constraints, specifying evolutionary relationships. Two canonical $NP$-hard optimization problems are Triplet and Quartet Reconstruction, where the input consists of triplets $xy|z$ or quartets $xy|zw$, and the goal is to find a tree $T^*$ maximizing agreement with constraints. Our main results are (as density $λ=m/n$ increases): 1. We show the existence and precisely locate the sharp threshold $λ^*\approx1.2277$ for Triplets (via closed-form solution). To the best of our knowledge, this is the first sharp threshold for the broad family of Phylogenetic CSPs. Moreover, we give a lower and upper bound for Quartets. 2. We provide strong refutation algorithms that certify that $val(T^*)\le5/9 + ε$, where $val(T^*)$ is the fraction of constraints satisfied by the (unknown) optimal tree. For triplets, our algorithm succeeds w.h.p if $m =Ω(n)$, and for quartets if $m = Ω(n^{3/2})$. 3. We obtain strongest possible refutations at slightly larger densities (for triplets $m=O(n^{3/2}\log ^3n)$, for quartets $m=O(n^2)$): we certify that $T^*$ is no better than a random assignment, i.e., $val(T^*)\le 1/3+ε$. In fact, we obtain strongest possible refutations for finite-alphabet CSPs with or without negations. Our refutations above are instantiations of our general theorem that applies more broadly to Phylogenetic and Ordering CSPs (and all CSPs failing to support $t$-wise independence), and generalizes the current algorithmic frontier on refuting random CSPs~\citep{allen2015refute}. A crucial difference here, unlike Boolean CSPs, is that there are no negated variables, so prior works relying on negations -- a source of randomness -- do not apply.

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