  The sequence of entropy numbers quantifies the degree of compactness of a
linear operator acting between quasi-Banach spaces. We determine the asymptotic
behavior of entropy numbers in the case of natural embeddings between
finite-dimensional Lorentz spaces $\ell_{p,q}^n$ in all regimes; our results
are sharp up to constants. This generalizes classical results obtained by
Sch\"utt (in the case of Banach spaces) and Edmunds and Triebel, K\"uhn, as
well as Gu\'edon and Litvak (in the case of quasi-Banach spaces) for entropy
numbers of identities between Lebesgue sequence spaces $\ell_p^n$. We employ
techniques such as interpolation, volume comparison as well as techniques from
sparse approximation and combinatorial arguments. Further, we characterize
entropy numbers of embeddings between symmetric quasi-Banach spaces satisfying
a lattice property in terms of best $s$-term approximation numbers.
