Estimates for trilinear and quadrilinear character sums

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to a modular analogue of a multiplicative hybrid problem of Iwaniec and S\'ark\"ozy (1987) is given. Moreover, we show that for all large primes p, there are primes p₁, p₂, p₃ of sizes p^ 15+ 11987 such that p₁p₂+p₃ is a quadratic non-residue modulo p, and there are also primes q₁, q₂, q₃, q₄ of sizes p^ 18+ 12024 such that q₁q₂+q₃ q₄ is a quadratic non-residue modulo p.