# Acids used in Friedel-Crafts alkylation

I am learning adding alkyl groups to benzene using Fridel-Crafts alkylation:

I know that it is very common to use $$\ce{AlCl3/AlBr3}$$ as Lewis acid catalyst in this reaction. But I am just wondering whether I can use a Bronsted acid, e.g., $$\ce{H2SO4}$$, as the catalyst or not. For example, in a $$\mathrm{E1}$$ or hydration reaction, $$\ce{H2SO4}$$ is added to alkene or alcohol, respectively, forming a carbocation, which is the same ultimate result when $$\ce{AlCl3}$$ is used (make the alkyl group more electrophilic).

I am thinking $$\ce{AlCl3}$$ is the only way if I am adding primary alkyl halide since in $$\mathrm{E1}$$/hydration reactions, an actual primary carbocation is impossible to be formed (too unstable), while it exists in a complex with $$\ce{AlCl3}$$ (not entirely a carbocation, so can be formed).

SO, what is the problem with secondary/tertiary alcohols if I use the Bronstad acid? If there is a problem, way? Both Lewis and Bronsted acid ways will lead to carbocation rearrangements (I have not learnt the acylation).