According to the classification of the quark model, the hadrons going beyond three-quark baryon and quark-antiquark meson pictures are called exotic hadrons. Many new hadrons have been observed since 2003, some of which exhibit exotic behaviors. There are a lot of excellent review articles on exotic hadrons available so far; the present article tries to focus on the recent experimental and theoretical progress on the exotic states from the perspective of the quark model. Although lattice quantum chromodynamics may give the final answer of the problem, the phenomenological models are still powerful tools to explore the exotic states and to provide insight on the phenomenology of hadrons. The spatial and color structures of multiquark states and the channel coupling calculation are emphasized through reviewing some bound states, molecular and color structure resonances. Finally, the unquench effects of some exotic states are reviewed. With the accumulation of experimental data on multiquark states and inspiration of underlying theory developments, more reasonable phenomenological models incorporating multi-body interactions and high Fock components to unify the description of normal hadrons and exotic hadrons are expectable.

Inspired by the observations of $T_{c\overline{s}0}^a(2900{)}^0$ and $T_{c\overline{s}0}^a(2900{)}^{++}$ by LHCb Collaboration, we perform a systematical investigation of the charmed-strange pentaquark system by using the resonance group method in the quark delocalization color screening model. The present estimations indicate the existence of two bound states with $I(J^P)$ to be $\frac{1}{2}({\frac{5}{2}}^{-})$ and $\frac{3}{2}({\frac{5}{2}}^{-})$, and the masses are predicted to be 3392 and 3307 MeV, respectively. Moreover, three resonance states are also predicted in the present work. The ${\mathrm{\Sigma }}_c{K}^{*}$ resonance with $I(J^P)=\frac{1}{2}({\frac{1}{2}}^{-})$ could be observed in the $N{D}_s^{*}$, ${\mathrm{\Lambda }}_c{K}^{*}$ and ${\mathrm{\Sigma }}_{c}K$ channels, the mass and width are estimated to be (3342–3346) MeV and 25.5 MeV, respectively. The ${\mathrm{\Sigma }}_c{K}^{*}$ resonance with $I(J^P)=\frac{1}{2}({\frac{3}{2}}^{-})$ is available only in the ${\mathrm{\Sigma }}_c^{*}K$ channel, and the resonance parameters are evaluated to be $m_R=3333\mathrm{MeV}$ and ${\mathrm{\Gamma }}_R=3.3\mathrm{MeV}$. We also find a very narrow $\mathrm{\Delta }{D}_s^{*}$ resonance state in the ${\mathrm{\Sigma }}_{c}K$ channel with $I(J^P)=\frac{3}{2}({\frac{1}{2}}^{-})$ and $m_R=3343\mathrm{MeV}$. These predictions of the charmed-strange pentaquark may be accessible by further experimental measurements in Belle II and LHCb Collaborations, and the corresponding experimental measurements can in turn be a good test of present estimations.

Inspired by the recent report from the LHCb Collaboration on $T_{cc}$, which can be interpreted as a molecular $D{D}^{*}$, we investigated two trimeson systems of the $T_{cc}$ partner with $I{J}^P=01^{-}$ in the framework of a chiral quark model. In the first case, because of the attraction between $D^{*}$ and $\overline{D}$, we explored the existence of bound states in the system $D^{*}D\overline{D}$, which is obtained by adding $\overline{D}$ into the molecular bound state $T_{cc}$ ($D{D}^{*}$). Similarly, in the second case, we explored whether there is a bound state in the $D^{*}DK$ system, which is obtained by adding $K$ into the $T_{cc}$, given the attraction between $D$ and $K$. The results show that both of them are bound states, in which the binding energy of the molecular state $D{D}^{*}K$ is relatively small, only $0.4\pm 0.4\mathrm{MeV}$, while the binding energy of $D{D}^{*}\overline{D}$ is up to $1.6\pm 0.3\mathrm{MeV}$. According to the calculation results of the root-square-mean distances, the spatial structure of the two systems shows the obvious $(D{D}^{*})\text{−}(\overline{D}/K)$ structure, in which $D$ is close to $D^{*}$ while $D{D}^{*}$ as a whole is relatively distant from the third hadron ($\overline{D}/K$), which are similar to the nucleon-electron structure. As a result, we strongly recommend that these bound states $D{D}^{*}\overline{D}$ and $D{D}^{*}K$ are searched for experimentally.

This study investigates the resonance $\omega {\chi }_{bJ}$, recently discovered in $e^{+}{e}^{-}$ collisions with a center-of-mass energy of $\sqrt{s}=10.745\mathrm{GeV}$ and reported by the Belle II Collaboration, to explore its potential identification with the previously reported $\mathrm{\Upsilon }(10753)$. We consider both the traditional $\mathrm{\Upsilon }(nS)$ meson and the exotic tetraquark $b\overline{q}q\overline{b}$ state with $J^P=1^{-}$, employing the chiral quark model to solve the Schrödinger equation. Our calculations demonstrate that the mass of $\mathrm{\Upsilon }(5S)$, $\mathrm{\Upsilon }(4S)$, and $\mathrm{\Upsilon }(3D)$, as potential two-quark state candidates, are inconsistent with the experimentally observed $\mathrm{\Upsilon }(10753)$, effectively excluding the possibility of $\mathrm{\Upsilon }(10753)$ as a two-quark state in our model. Furthermore, we employ the Gaussian expansion method to investigate the tetraquark structure of the resonance, including two molecular structures ($b\overline{b}\text{−}q\overline{q}$, $b\overline{q}\text{−}q\overline{b}$) and a diquark-antidiquark structure ($\overline{b}\overline{q}\text{−}qb$), and perform a full-channel coupling using the real-scaling method. Our investigation yields a total of seven resonant states, with one state having an energy of $10761\pm 7\mathrm{MeV}$ and a width of $10.8\pm 5.2\mathrm{MeV}$, which is a promising candidate for the experimentally observed $\mathrm{\Upsilon }(10753)$. Furthermore, we report no resonances in the energy range below 10.6 GeV, while several resonances were found in the 10.8–10.9 GeV range. These findings suggest the need for future experiments to search for additional resonant states in this energy range, potentially providing further support for our model.

In the present work, the triply heavy tetraquarks states $QQ\overline{Q}\overline{q}$ with $Q=(c,b)$ and $q=(u,d,s)$ with all possible quantum numbers are systematically investigated in the framework of the chiral quark model with the resonating ground method. Two kinds of structures, including the meson-meson configuration (the color-singlet channels and the hidden-color channels) and the diquark-antidiquark configuration (the color sextet-antisextet and the color triplet-antitriplet), are considered. In the considered system, several bound states are obtained for the $cc\overline{c}\overline{q^{\prime }}$, $bb\overline{c}\overline{q^{\prime }}$, and $bc\overline{c}\overline{q}$ tetraquarks. From the present estimations, we find that the coupled-channel effect is of great significance for forming below-threshold tetraquark states, which are stable for strong decays.

Since the quark model was put forward, theoretical researchers have always attached great importance to the study of hidden color channels (including color octets and diquark structure). Because of the influence of color van der Waals forces, the hidden color channel itself has strong attraction, which provides a dynamic mechanism for the formation of a resonance state or bound state. In this paper, taking the $T_{cc}$ system as an example, under the framework of the Gaussian expansion method, a set of relatively complete color singlets (that is, the ground state of the color singlet plus its corresponding higher-order component) is used to replace the contribution of the color octet. Similarly, we endeavor to replace the diquark structure with a relatively complete set of molecular states, encompassing both the ground state and excited states. Our results demonstrate that the color-octet structure can be effectively replaced by a set of relatively complete color-singlet bases, while the diquark structure cannot be entirely substituted by an equivalently comprehensive set of molecular state bases.