Derivations and quasi-derivations of Block Lie superalgebras $S(q)$

We compute the homogeneous derivations and the homogeneous quasi-derivations of the Block Lie superalgebras $S(q)$, whose even part is the Block Lie algebra $\mathcal{B}(q)$. For every $q\in \mathbb{C}$, the space ${\mathfrak{D}\mathfrak{e}\mathfrak{r}}_{\theta }(S(q))$ is the direct sum of the inner derivations and an explicitly given finite-dimensional space of outer derivations; in particular, for $q\mathrm{\ne }0$ a new odd outer derivation appears if and only if $q\in 2{\mathbb{Z}}^{\ast }$. For $q\in {\mathbb{C}}^{\ast }$ we prove that $\mathfrak{Q}\mathfrak{D}\mathfrak{e}\mathfrak{r}(S(q))=\mathfrak{D}\mathfrak{e}\mathfrak{r}(S(q))\oplus ⟨\text{id}⟩$ if $q\mathrm{\notin }\mathbb{Z}$, and that for $q\in {\mathbb{Z}}^{\ast }$ the superalgebra $S(q)$ has an infinite-dimensional space of nontrivial quasi-derivations, namely all graded linear maps with values in the center. For $q=0$ we also describe all homogeneous quasi-derivations: modulo maps with values in the center, they are spanned by $\text{id}$, three further maps and infinite families of maps shifting the ${\mathbb{Z}}^2$-grading; in particular, $\mathfrak{Q}\mathfrak{D}\mathfrak{e}\mathfrak{r}(S(0))\mathrm{/}\mathfrak{D}\mathfrak{e}\mathfrak{r}(S(0))$ is infinite-dimensional. As corollaries, we obtain the complete description of homogeneous $\delta$-derivations of $S(q)$, $q\mathrm{\ne }0$, and prove that, in contrast with the Block Lie algebras $\mathcal{B}(q)$, the superalgebras $S(q)$, $q\mathrm{\ne }0$, do not admit nontrivial transposed $\delta$-Poisson superalgebra structures for any $\delta \mathrm{\notin }\{0,1\}$.