Statistical Inference — correlation of UMVUE with an unbiased estimator

Consider the following statements regarding unbiased estimators of parameter $\theta$: 1. Let $T_0$ be an UMVUE of $\theta$ and $T_1$ is any other unbiased estimator of $\theta$ with efficiency $0.64$, then the correlation coefficient between $T_0$ and $T_1$ is $0.8$. 2. Let $T_0$ be an MVUE of $g(\theta)$ and $T_1$ is any other unbiased estimator with efficiency less than 1, then any unbiased linear combination of $T_0$ and $T_1$ will also be MVUE of $g(\theta)$. Which of the above statements is/are correct?