  An upper bound on the feedback capacity of unifilar finite-state channels
(FSCs) is derived. A new technique, called the $Q$-contexts, is based on a
construction of a directed graph that is used to quantize recursively the
receiver's output sequences to a finite set of contexts. For any choice of
$Q$-graph, the feedback capacity is bounded by a single-letter expression,
$C_\text{fb}\leq \sup I(X,S;Y|Q)$, where the supremum is over $P_{X|S,Q}$ and
the distribution of $(S,Q)$ is their stationary distribution. It is shown that
the bound is tight for all unifilar FSCs where feedback capacity is known:
channels where the state is a function of the outputs, the trapdoor channel,
Ising channels, the no-consecutive-ones input-constrained erasure channel and
for the memoryless channel. Its efficiency is also demonstrated by deriving a
new capacity result for the dicode erasure channel (DEC); the upper bound is
obtained directly from the above expression and its tightness is concluded with
a general sufficient condition on the optimality of the upper bound. This
sufficient condition is based on a fixed point principle of the BCJR equation
and, indeed, formulated as a simple lower bound on feedback capacity of
unifilar FSCs for arbitrary $Q$-graphs. This upper bound indicates that a
single-letter expression might exist for the capacity of finite-state channels
with or without feedback based on a construction of auxiliary random variable
with specified structure, such as $Q$-graph, and not with i.i.d distribution.
The upper bound also serves as a non-trivial bound on the capacity of channels
without feedback, a problem that is still open.
