Auto-correlation function, an inverse problem $x[n]$ is a complex function $n=0,1,2,\cdots,L-1 $
we assume $x[n]$ is periodic in its index:  $x[n+L]=x[n]$
Its auto-correlation function $C[n]$ is uniquely defined as:
$$
C[n]=\sum_{i=0}^{L-1} x[i+n]x^*[i]
$$
$C[n]$ also has the periodic property: $$C[n+L]=C[n]\tag{1}$$
And ''conjugate-symmetry'' property: $C[-n]=C^*[n] \tag{2}$

Now my question is:
For given $C[n]$, which satisfies property (1) and (2):
Can we find the corresponding $x[n]$ ?
If yes, is it unique? 
$\qquad $ if unique, what is the method to find $x[n]$?
$\qquad $ if not unique, what is the class of those $C[n] \rightarrow \{x[n]\}$
If no, what other constraint properties should we add to $C[n]$, in order to make it yes?


Even up to shifts it is not unique at all. For example there is a whole collection of sequences called $m-$sequences (maximal length sequences) generated by binary linear shift registers corresponding to primitive polynomials. See the discussion on wikipedia.
There are $\phi(2^n-1)/n$ different primitive polynomials over the binary field of degree $n$ and each of these give rise to an $m-$sequence (and all its shifts). All these sequences have period $2^n-1$ and ideal autocorrelation
$$C_t=-1+\delta(t)2^n$$
where $\delta(\cdot)$ is the Kronecker delta.
Corresponding complex valued sequences exist for non-binary fields, over $GF(p)$ they are sequences over complex roots of unity of order $p.$
