Motivated by high-temperature superconductivity in pressurized La$_3$Ni$_2$O$_7$, we investigate the pair correlations in the two-orbital Hubbard ladder, which consists of the nearly half-filled and nearly quarter-filled orbitals. By employing the density matrix renormalization group method, we demonstrate that the pair correlation exhibits a power-law decay against the distance while the spin correlation decays exponentially. The decay exponent of the pair correlation of the nearly half-filled orbital is comparable to the exponent of the quasi-long-range superconducting correlation in the doped single-orbital Hubbard ladder, which suggests the importance of the $d_{3z^2-r^2}$ orbital in La$_3$Ni$_2$O$_7$.

We theoretically investigate the electronic structure of monolayer BC$_3$ and find that it hosts anisotropic multiple valleys originating from the splitting of the van Hove singularity in graphene. To make use of its favorable electronic structure, we investigate the electronic structure of alkali-metal-intercalated BC$_3$, where intercalated atoms not only introduce electron carriers but also suppress interlayer coupling. We find that the interlayer transfer is effectively suppressed by potassium intercalation, by which the favorable electronic structure of monolayer BC$_3$ is preserved. Finally, we perform model calculation with the onsite-energy offset, and we verify that the strategy of introducing the splitting to the van Hove singularity works well.