Motivated by the problem of determining the atomic structure of macromolecules using single-particle cryo-electron microscopy (cryo-EM), we study the sample and computational complexities of the sparse multi-reference alignment (MRA) model: the problem of estimating a sparse signal from its noisy, circularly shifted copies. Based on its tight connection to the crystallographic phase retrieval problem, we establish that if the number of observations is proportional to the square of the variance of the noise, then the sparse MRA problem is statistically feasible for sufficiently sparse signals. To investigate its computational hardness, we consider three types of computational frameworks: projection-based algorithms, bispectrum inversion, and convex relaxations. We show that a state-of-the-art projection-based algorithm achieves the optimal estimation rate, but its computational complexity is exponential in the sparsity level. The bispectrum framework provides a statistical-computational trade-off: it requires more observations (so its estimation rate is suboptimal), but its computational complexity is provably polynomial in the signal's length. The convex relaxation approach provides polynomial-time algorithms (with a large exponent) that recover sufficiently sparse signals at the optimal estimation rate. We conclude the paper by discussing potential statistical and algorithmic implications for cryo-EM.