Quantum mechanical frustration (QMF) in magnetic materials has become a pivotal ingredient in discovering intriguing properties of materials. The quantum spin liquid (QSL) state is a prime consequence of frustration in which spin fluctuations persist to absolute zero temperature. This orderless state is not characterized by symmetry breaking and guarantees an infinite degeneracy in its ground state. The quest to realize such nontrivial ground states in view of spin correlation, elementary excitation, topology, and geometry requires convincing experimental evidence. QMF is often manifested in unique lattice systems, such as spin-1/2 hyper-honeycomb lattices with strong spin-orbit coupling and geometrically frustrated Kagome lattices. Metal-organic frameworks (MOFs) that comprise metal ions with organic linkers via coordinate bonds have recently been proposed to realize the QSL ground state. Owing to the vast versatility of constituent's, the copper-oxalate MOF, [(C2H5)3NH]2Cu2(C2O4)3, unfolds a new avenue for us to realize unusual magnetic phases. To investigate the exotic ground state, we synthesized single crystals of [(C2H5)3NH]2Cu2(C2O4)3 and measured their thermodynamic properties. Our low-temperature and high-magnetic-field heat-capacity (Cp) measurements corroborate an exotic but rich ground state in [(C2H5)3NH]2Cu2(C2O4)3. A finite linear-in temperature Cp term with no indication of any thermal anomaly was observed at low temperature in zero field, indicating the absence of magnetic order and the presence of gapless spinon excitations despite the Mott insulating phase. Applied magnetic fields suppress the low-temperature Cp and drive the system into a gapped phase. The field-induced gap is described by the sine-Gorden model for quasi-one-dimensional antiferromagnetic Heisenberg chains, originating from anisotropic magnetic exchange interactions due to the Jahn-Teller distortion. Kagome lattices are archetypes of potential QSL, superconducting, Chern insulating states due to flat energy bands, Dirac fermions, and van Hove singularities in its electronic band structure. Sc3Mn3Al7Si5 is a transition metal compound with a hexagonal structure in which magnetic Mn atoms form kagome nets and does not show magnetic order down to 2 K, suggestive of a possible itinerant QSL. In this dissertation, to elucidate its exotic ground state, we synthesized single crystals of Sc3Mn3Al7Si5 and measured magnetoresistance, Cp, soft point contact spectroscopy, and torque magnetometry at low temperatures and high magnetic fields. Our experimental results suggest that the unusual ground state is induced by dispersionless energy bands induced by strong electron correlations. Our findings through distinct states of matter spanning from Mott insulators to itinerant metals will provide new insights to characterize the ground state in novel quantum magnets.