Abstract The main purpose of this paper is to study prime spectrums of EQ-algebras and to solve two open problems about $\wedge $-prime spectrums of involutive and prelinear EQ-algebras, which were proposed by N. Akhlaghinia, R.A. Borzooei and M. A. Kologani. In order to do so, we first give some characterizations of preideals, prime preideals and maximal preideals on (good) EQ-algebras, respectively. Then we introduce the notion of quasi De Morgan EQ-algebras (MEQ-algebras for short) and obtain that $\wedge $-prime preideals coincide with prime preideals for MEQ-algebras, and each involutive EQ-algebra is an MEQ-algebra. Following, we show that the prime spectrum space of a good EQ-algebra is a compact topological space and obtain that for any involutive EQ-algebra the prime spectrum space is connected if and only if its Boolean center is indeed 2-element. Also, we prove that the maximal spectrum space of a good and prelinear EQ-algebra (or an involutive and prelinear EQ-algebra) is a normal Hausdorff space. These results totally answer the above two open problems. Finally, we give some characterizations of the spectrum space of an MEQ-algebra by its reticulation.