Simulate time series given temporal auto-correlation functions
Here is one possible method that you could possibly use.

Let's model your random process $x[n]$ as an auto-regressive (AR) process of order $P$, that is,
$$x[n] + \sum_{p=1}^P a_px[n-p] = c + \epsilon[n]$$
where $c$ is some constant that I will set to 0 in what follows for the sake of simplicity, $\{a_p\}_{p=1}^P$ are the parameters of the AR model and $\epsilon[n]$ is some white noise of variance $\sigma^2_\epsilon$.

Multiplying each side of the above equation by $x[n-l]$ for some $l \ge 0$ leads us to
$$\left(x[n] + \sum_{p=1}^P a_px[n-p]\right)x[n-l] = \epsilon[n]x[n-l].$$
Now, taking the expectation on each side, we obtain the Yule-Walker equations
$$C_x[l] + \sum_{p=1}^P a_pC_x[l-p] = 
\begin{cases}
0 &amp; \text{if} &amp; l &gt; 0, \\
\sigma^2_\epsilon &amp; \text{if} &amp; l = 0
\end{cases}$$
using the fact that $\epsilon[n]$ and $x[n-l]$ are uncorrelated for $l &gt; 0$ (you can see that from the equation of the AR process above). Using this equation with $l = 0, 1, 2, \dots, P$, you have $P+1$ equations to solve for the $P + 1$ unknowns (i.e., $\{a_p\}_{p=1}^P$ and $\sigma^2_\epsilon$). 

You then have a model to generate a time series given the temporal auto-correlation.