  A number of spectacular experimental anomalies\cite{li-2007,fujita-2005} have
recently been discovered in certain cuprates, notably {\LBCO} and {\LNSCO},
which exhibit unidirectional spin and charge order (known as ``stripe order'').
We have recently proposed to interpret these observations as evidence for a
novel ``striped superconducting'' state, in which the superconducting order
parameter is modulated in space, such that its average is precisely zero. Here,
we show that thermal melting of the striped superconducting state can lead to a
number of unusual phases, of which the most novel is a charge $4e$
superconducting state, with a corresponding fractional flux quantum $hc/4e$.
These are never-before observed states of matter, and ones, moreover, that
cannot arise from the conventional Bardeen-Cooper-Schrieffer (BCS) mechanism.
Thus, direct confirmation of their existence, even in a small subset of the
cuprates, could have much broader implications for our understanding of high
temperature superconductivity. We propose experiments to observe fractional
flux quantization, which thereby could confirm the existence of these states.
