the frustrated quantum spin model on a square lattice that contains bilinear and biquadratic interactions. For spin $S=1$, using semi-classical site-factorized wavefunction analysis and fully-quantum density matrix renormalization group (DMRG) studies, we provide evidence for a phase regime in which the BC order is degenerate with the PL order. The degeneracy is robust, persisting even for the higher $S=3/2, 2$ systems as found by the DMRG calculations. We then determine the fluctuation spectra of the BC and PL phases using a flavor-wave theory analysis for the spin-$1$ case. This allows us to identify a regime where quantum fluctuations select the BC order, by destabilizing the PL order. For the much more extended parameter regime where both the BC and PL states are stable and degenerate, we show that a nonzero $4$-site ring-exchange interaction selects the BC order.

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*Model.*— We study the $J$-$K$-$\mathfrak{R}$ model whose Hamiltonian is defined as $$\begin{aligned}
 H&=&  \sum_{i, j} \left[ J_{ij} {\bf S}_i \cdot {\bf S}_j + K_{ij} \left( {\bf S}_i \cdot {\bf S}_j \right)^2 \right] \nonumber \\ 
 &-& \sum_{\triangpic}  \mathfrak{R}_3 \left( P_{123} + \Hc \right) + \sum_{\rhombpic} \mathfrak{R}_4  \left( P_{1234} + \Hc \right), 
\label{ham_ring}~~~~~\end{aligned}$$ where ${\bf S}_{i}$ is the local moment at site $i$ of 