This paper studies the large time behavior of solutions to semilinear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: (i) pointwise convergence of the solution and its gradient and (ii) convergence of solutions to associated backward stochastic differential equations. When the state space is R-d or the space of positive definite matrices, both types of convergence are obtained under growth conditions on coefficients. These large time convergence results have direct applications in risk-sensitive control and long-term portfolio choice problems.