Acids can differentiate $\ce{Al2O3}$ and $\ce{SiO2}$. There is a famous reaction called etching of glass where $\ce{SiO2}$ reacts with hydrofluoric acid $\ce{HF}$. $\ce{Al2O3}$ also reacts with $\ce{HF}$ but it requires heat. This test somehow differentiates them but we need to determine which substance is which by reacting with an acid which reacts with only one oxide. Acids like $\ce{HCl, H2SO4}$ will suffice.

$$\ce{Al2O3 + 3H2SO4 ->[\Delta] Al2(SO4)3 + 3H2O}$$

(Here)

Silicon(IV) oxide does not react with sulfuric acid rather silicon(II) oxide reacts with sulfuric acid to produce silicon(IV) oxide.

$$\ce{Al2O3 + 6HCl ->[\Delta] 2AlCl3 + 3H2O}$$

Here

$$\ce{Al2O3 + 6HCl + 9H2O → 2 [Al (H2O)6]Cl3}$$

(Here)

$\ce{SiO2}$ does not react with hydrochloric acid.

Acids can differentiate $\ce{Al2O3}$ and $\ce{SiO2}$. There is a famous reaction called etching of glass where $\ce{SiO2}$ reacts with hydrofluoric acid $\ce{HF}$. $\ce{Al2O3}$ also reacts with $\ce{HF}$ but it requires heat. This test somehow differentiates them but we need to determine which substance is which by reacting with an acid which reacts with only one oxide. Strong acids like $\ce{HCl, H2SO4}$ or will suffice.

It reacts with acids to form corresponding aluminium salts(sulfate, chloride)

$$\ce{Al2O3 + 3H2SO4 ->[\Delta] Al2(SO4)3 + 3H2O}$$

$$\ce{Al2O3 + 6HCl ->[\Delta] 2AlCl3 + 3H2O}$$

Silicon(IV) oxide does not react with any of the above acid. Rather silicon(II) oxide reacts with sulfuric acid or nitric acid to produce silicon(IV) oxide.