For \(K \subset Y\), where \(Y\) is a finite-dimensional (quasi-)Banach space and \(B_{Y}\) denotes the closed unit ball in \(Y\), the \(k\)th (dyadic) entropy number \(e_{k}(K,Y)\) is defined as \[ e_{k}(K,Y) = \min \Bigg\{ \varepsilon > 0 : K \subset \bigcup_{j=1}^{2^{k-1}} (x_{j}+\varepsilon B_{Y}) \text{ for some } x_{1},\dots,x_{2^{k-1}} \in Y \Bigg\}. \] The authors prove sharp upper bounds on the entropy numbers \(e_{k}(\mathbb{S}_{p}^{d-1},\ell_{q}^{d})\) of the \(p\)-spheres \(\mathbb{S}_{p}^{d-1}=\{ x \in \mathbb{R}^{d} : \| x \|_{p} = 1 \}\) in \(\ell_{q}^{d}=(\mathbb{R}^{d},\| \cdot \|_{q})\) in the case \(k \geq d\) and \(0<p\leq q \leq \infty\) (the (quasi-)norms are defined by \(\| x \|_{p} := (\sum_{i=1}^{d} \, | x_{i} |^{p})^{1/p}\) for \(0<p<\infty\) and \(\| x \|_{\infty} := \max \{ | x_{i} | : i=1,2,\dots,d \}\)). Finally, the presented methods are generalized in order to study entropy numbers \(e_{k}(\mathbb{S}_{X},Y)\) for general finite-dimensional quasi-Banach spaces \(X\) and \(Y\).