Recently, I am considering a class of character sums concerning Legendre symbols. Let $p$ be an odd prime, and $\phi$ the Legendre symbol mod $p$. It is well known that $$\sum_x\phi(x+a)\phi(x+b)=-1,\ \ (a-b,p)=1,$$where $x$ runs over the residue system mod $p$. However, I haven't found any references on the character sums $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)$$ and $$\sum_x\phi(x+a)\phi(x+b)\phi(x+c)\phi(x+d).$$

Of courese, the sums of such types are useful as studying the consecutive quadratic residues.