The initial question comes from Komal in 1999.

Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are distinct powers of $2$. This is equivalent to finding $(t_0,\dotsc,t_n)$ different positive integers such that $\displaystyle \sum_{i=0}^{n} (-1)^{n-i}\dbinom{n+1}{i} 2^{t_i}$ is a power of $2$.

We could ask something more: Is it true that there exists $c_n$ and we can bound it in terms of $n$ such that for all $f\in\mathbb{Q}[X]$ of degree $n$ we have that $f(0),f(1),\dotsc, f(c_n)$ cannot all be powers of $2$? The existence of $c_n$ and that it is bounded in terms of $n$ follows from a strengthened conjectural version of Falting's theorem for curves of the type $y^m=f(x)$. Can we say something unconjecturally about this? For $f(0),f(1),f(2),\dotsc, f(n)$ distinct powers of $2$ we can even construct $f\in\mathbb{Z}[X]$ thus $c_n\geq n+1$ always.