The goal of this talk is to consider two instances of a class of reconstruction problems that aim to recover an unknown signal from indirect measurements that are algebraic in nature. Such problems are paramount in mathematics, enjoying applications in a wide array of fields like molecular imaging, machine learning, and geo-positioning. In this talk, we will motivate and study the generic crystallographic phase retrieval problem and the orthogonal beltway problem and deduce conditions in each setting that guarantee signal recovery. In the latter problem, we will also develop a polynomial-time algorithm that recovers the signal while remaining robust to small amounts of noise. We resolve both of the central problems of this talk by recasting them as special instances of the problem of recovering a signal from its second moment under the multi-reference alignment (MRA) model, for which a rich theory has been developed in prior literature.