The correct order of increasing thermal stability of ${\text{K}}_2{\text{CO}}_3$, ${\text{MgCO}}_3$, ${\text{CaCO}}_3$ and ${\text{BeCO}}_3$ is:

1. For alkaline earth metal carbonates ($\text{Be}$, $\text{Mg}$, $\text{Ca}$), thermal stability increases down the group as the electropositive character increases and the polarizing power of the cation decreases. So: ${\text{BeCO}}_3<{\text{MgCO}}_3<{\text{CaCO}}_3$.2. Alkali metal carbonates (except ${\text{Li}}_2{\text{CO}}_3$) are more stable than alkaline earth metal carbonates because alkali metals are more electropositive. ${\text{K}}_2{\text{CO}}_3$ is extremely stable and does not decompose easily.Therefore, the order is: ${\text{BeCO}}_3<{\text{MgCO}}_3<{\text{CaCO}}_3<{\text{K}}_2{\text{CO}}_3$.