  We study higher uniformity properties of the M\"obius function $\mu$, the von
Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals
$(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a
fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$.
  More precisely, letting $\Lambda^\sharp$ and $d_k^\sharp$ be suitable
approximants of $\Lambda$ and $d_k$ and $\mu^\sharp = 0$, we show for instance
that, for any nilsequence $F(g(n)\Gamma)$, we have \[ \sum_{X < n \leq X+H}
(f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \] when $\theta = 5/8$ and
$f \in \{\Lambda, \mu, d_k\}$ or $\theta = 1/3$ and $f = d_2$.
  As a consequence, we show that the short interval Gowers norms
$\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$
for these choices of $f,\theta$. As applications, we prove an asymptotic
formula for the number of solutions to linear equations in primes in short
intervals, and show that multiple ergodic averages along primes in short
intervals converge in $L^2$.
  Our innovations include the use of multi-parameter nilsequence
equidistribution theorems to control type $II$ sums, and an elementary
decomposition of the neighbourhood of a hyperbola into arithmetic progressions
to control type $I_2$ sums.
