  These notes survey and explore an emerging method, which we call the
low-degree method, for predicting and understanding
statistical-versus-computational tradeoffs in high-dimensional inference
problems. In short, the method posits that a certain quantity -- the second
moment of the low-degree likelihood ratio -- gives insight into how much
computational time is required to solve a given hypothesis testing problem,
which can in turn be used to predict the computational hardness of a variety of
statistical inference tasks. While this method originated in the study of the
sum-of-squares (SoS) hierarchy of convex programs, we present a self-contained
introduction that does not require knowledge of SoS. In addition to showing how
to carry out predictions using the method, we include a discussion
investigating both rigorous and conjectural consequences of these predictions.
  These notes include some new results, simplified proofs, and refined
conjectures. For instance, we point out a formal connection between spectral
methods and the low-degree likelihood ratio, and we give a sharp low-degree
lower bound against subexponential-time algorithms for tensor PCA.
