At first I describe the connection of the QR algorithm with the power method and the inverse power method. As I understand this is the main topic you are interested in. Later -- when I have a little more time -- I will extent this to subspace iteration. I think that is what the second part of your question is about.
Keeping the discussion simple, let $A\in\mathbb{C}^{n\times n}$ be a regular complex square matrix
with eigenvalues having pairwise distinct absolute values.
The QR algorithm without shift is defined by the iteration
\begin{align*}
&\text{Start}&A_1 &:= A\\
&\text{QR-decomposition}& Q_i R_i &:= A_i &&@i=1,\ldots\\
&\text{rearranged new iterate}&A_{i+1} &:= R_i Q_i
\end{align*}
Representing $R_i$ as
$R_i = Q_i^H A_i$ and substituting this into the formula for $A_{i+1}$ gives
$ A_{i+1} = Q_i^H A_i Q_i$. Thus, the matrix $A_{i+1}$ is similar to $A_i$ and has the same eigenvalues.
Defining the combined orthogonal transformation $\bar Q_i := Q_1\cdot\ldots\cdot Q_i$ for $i=1,\ldots$ we obtain
\begin{align*}
A_{i+1} &= \bar Q_i^H A \bar Q_i&&@ i=1,\ldots
\end{align*}
or $A_i = \bar Q^H_{i-1} A \bar Q_{i-1}$ for $i=2,\ldots$. We substitute $A_i$ in the above QR-decomposition with this formula and obtain
\begin{align*}
Q_i R_i &= A_i = \bar Q_{i-1}^H A \bar Q_{i-1}\\
\bar Q_i R_i &= A \bar Q_{i-1},
\end{align*}
using the definition of $\bar Q_{i-1}Q_i = \bar Q_i$ in the second equality.
Taking the first column on both sides of the last equation one gets
\begin{align*}
\bar Q_i(:,1)\cdot R_i(1,1) = A\cdot \bar Q_{i-1}(:,1)
\end{align*}
(with Octave notation of the matrix elements) which shows that
$R_i(1,1)$ and $\bar Q_i(:,1)$ are just the approximations for the
eigenvalue and the eigenvector, respectively, in the power-iteration
for the matrix $A$. The elements $R_i(1,1)$ with $i=1,\ldots$ converge to largest of the
eigenvalues whose eigenvectors belong to the invariant subspace of $A$
spanned by the vectors $A^k Q_1(:,1)$ with $k=0,\ldots,n-1$.
Analogously to the above procedure we use the formula for the
rearranged new iterate of the QR-algorithm to get
\begin{align*}
\bar Q_i^H A \bar Q_i &= A_{i+1} = R_i Q_i\\
\bar Q_i^H A &= R_i \bar Q_{i-1}^H
\end{align*}
The last row of the last equation results to
\begin{align*}
\bar Q_i^H(n,:) \cdot A &= R_i(n,n)\cdot \bar Q^H_{i-1}(n,:).
\end{align*}
Theoretically, one can calculate the sequences $\bar Q_i^H(n,:)$ and
$R_i(n,n)$ for $i=2,\ldots$ from $\bar Q_1^H(n,:)=Q_1^H(n,:)$ by the
inverse power iteration
\begin{align*}
&\text{Solve}& x\cdot A &= \bar Q_{i-1}^H(n,:)&&\text{for }x\in\mathbb{C}^{1,n}\\
&\text{Normalize} & \bar Q_i^H(n,:) &:= \frac{x}{|x|}\\
&\text{Set}& R_i(n,n) &:= |x|
\end{align*}
where $R_i(n,n)$ are the approximations for the smallest of the eigenvalues of
$A$ with corresponding left-eigenvector in the subspace spanned by the vectors $Q_1^H(n,:)\cdot A^k$ with $k=0,\ldots,n-1$.
