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The magnetic and quadrupole moments of the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$and $Z_{c}(4600)^{+}$ states are calculated within the QCD light-cone sumrules. The compact diquark-antidiquark interpolating currents and thedistribution amplitudes of the on-shell photon are used to extract the magneticand quadrupole moments of these states. The magnetic moments are acquired as$\mu_{Z_{c}} = 0.50 ^{+0.22}_{-0.22}~\mu_N$, $\mu_{Z^{1}_{c}}=1.22^{+0.34}_{-0.32}~\mu_N$, and $\mu_{Z^2_{c}}=2.40 ^{+0.53}_{-0.48}~\mu_N$ forthe $Z_{c}(4020)^+$, $Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states,respectively. The magnetic moments evaluated for the $Z_{c}4020)^+$,$Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states are sufficiently large to beexperimentally measurable. The magnetic moment is an excellent platform forstudying the internal structure of hadrons governed by the quark-gluon dynamicsof QCD because it is the leading-order response of a bound system to a weakexternal magnetic field. The quadrupole moment results are$\mathcal{D}_{Z_c}=(0.20 ^{+0.05}_{-0.04}) \times 10^{-3}~\mbox{fm}^2 $,$\mathcal{D}_{Z_c^1}=(0.57 ^{+0.07}_{-0.08}) \times 10^{-3}~\mbox{fm}^2 $, and$\mathcal{D}_{Z_c^2}=(0.30 ^{+0.05}_{-0.04}) \times 10^{-3}~\mbox{fm}^2 $ forthe $Z_{c}(4020)^+$, $Z_{c}(4050)^+$ and $Z_{c}(4600)^{+}$ states,respectively. We obtain a non-zero, but small, value for the quadrupole momentsof the $Z_c$ states, which indicates a non-spherical charge distribution. Thenature and internal structure of these states can be elucidated by comparingfuture experimental data on the magnetic and quadrupole moments of the$Z_{c}(4020)^+$, $Z_{c}(4050)^+$, and $Z_{c}(4600)^{+}$ states with the resultsof the present study.

Magnetic and quadrupole moments of the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$, and $Z_{c}(4600)^{+}$ states in the diquark-antidiquark picture