We present a generalization of the hyperspherical harmonic formalism to study systems made of quarks and antiquarks of the same flavor. This generalization is based on the symmetrization of the $N$-body wave function with respect to the symmetric group using the Barnea and Novoselsky algorithm. The formalism is applied to study four-quark systems by means of a constituent quark model successful in the description of the two- and three-quark systems. The results are compared to those obtained by means of variational approaches. Our analysis shows that four-quark systems with exotic $0^{+-}$ and nonexotic $2^{++}$ quantum numbers may be bound independently of the mass of the quark. $2^{+-}$ and $1^{+-}$ states become attractive only for larger mass of the quarks.

We present an exact calculation of $S$ and $P$ wave $QQ\overline{n}\overline{n}$ states using different standard nonrelativistic quark-quark potentials. We explore in detail the charm and bottom sectors looking for bound states that could be measured within existing facilities. Against the proliferation of four-quark states sometimes predicted in the literature, we found a small number of candidates to be stable. We analyze their properties in a trial to distinguish between compact and molecular states. Possible decay modes are discussed.